(In the past, the editorial boards of several Elsevier journals have resigned over the pricing policies of the journal, most famously the board of Topology in 2006, but also the Journal of Algorithms in 2003, and a number of journals in other sciences as well.  Several libraries, such as those of Harvard and Cornell, have also managed to negotiate an unbundling of Elsevier journals, but most libraries are still unable to subscribe to such journals individually.)

For a more thorough discussion as to why such a protest is warranted, please see Tim’s post on the matter (and the 100+ comments to that post).   Many of the issues regarding Elsevier were already known to some extent to many mathematicians (particularly those who have served on departmental library committees), several of whom had already privately made the decision to boycott Elsevier; but nevertheless it is important to bring these issues out into the open, to make them commonly known as opposed to merely mutually known.  (Amusingly, this distinction is also of crucial importance in my favorite logic puzzle, but that’s another story.)   One can also see Elsevier’s side of the story in this response to Tim’s post by David Clark (the Senior Vice President for Physical Sciences at Elsevier).

For my own part, though I have sent about 9% of my papers in the past to Elsevier journals (with one or two still in press), I have now elected not to submit any further papers to these journals, nor to serve on their editorial boards, though I will continue refereeing some papers from these journals.  As of this time of writing, over five hundred mathematicians and other academics have also signed on to the protest in the four days that the site has been active.

Admittedly, I am fortunate enough to be at a stage of career in which I am not pressured to publish in a very specific set of journals, and as such, I am not making a recommendation as to what anyone else should do or not do regarding this protest.  However, I do feel that it is worth spreading awareness, at least, of the fact that such protests exist (and some additional petitions on related issues can be found at the previously mentioned wiki page).

If we normalise the entries of the matrix to have mean zero and variance , then in the asymptotic limit , we have the Wigner semicircle law, which asserts that the eigenvalues are asymptotically distributed according to the semicircular distribution , where

An essentially equivalent way of saying this is that for large , we expect the eigenvalue of to stay close to the classical location , defined by the formula

In particular, from the Wigner semicircle law it can be shown that asymptotically almost surely, one has

for all .

In the modern study of the spectrum of Wigner matrices (and in particular as a key tool in establishing universality results), it has become of interest to improve the error term in (1) as much as possible. A typical early result in this direction was by Bai, who used the Stieltjes transform method to obtain polynomial convergence rates of the shape for some absolute constant ; see also the subsequent papers of Alon-Krivelevich-Vu and of of Meckes, who were able to obtain such convergence rates (with exponentially high probability) by using concentration of measure tools, such as Talagrand’s inequality. On the other hand, in the case of the GUE ensemble it is known (by this paper of Gustavsson) that has variance comparable to in the bulk, so that the optimal error term in (1) should be about . (One may think that if one wanted bounds on (1) that were uniform in , one would need to enlarge the error term further, but this does not appear to be the case, due to strong correlations between the ; note for instance this recent result of Ben Arous and Bourgarde that the largest gap between eigenvalues in the bulk is typically of order .)

A significant advance in this direction was achieved by Erdos, Schlein, and Yau in a series of http://www.ams.org/mathscinet-getitem?mr=2587574{papers} where they used a combination of Stieltjes transform and concentration of measure methods to obtain local semicircle laws which showed, among other things, that one had asymptotics of the form

with exponentially high probability for intervals in the bulk that were as short as for some , where is the number of eigenvalues. These asymptotics are consistent with a good error term in (1), and are already sufficient for many applications, but do not quite imply a strong concentration result for individual eigenvalues (basically because they do not preclude long-range or “secular” shifts in the spectrum that involve large blocks of eigenvalues at mesoscopic scales). Nevertheless, this was rectified in a subsequent paper of Erdos, Yau, and Yin, which roughly speaking obtained a bound of the form

in the bulk with exponentially high probability, for Wigner matrices obeying some exponential decay conditions on the entries. This was achieved by a rather delicate high moment calculation, in which the contribution of the diagonal entries of the resolvent (whose average forms the Stieltjes transform) was shown to mostly cancel each other out.

As the GUE computations show, this concentration result is sharp up to the quasilogarithmic factor . The main result of this paper is to improve the concentration result to one more in line with the GUE case, namely

with exponentially high probability (see the paper for a more precise statement of results). The one catch is that an additional hypothesis is required, namely that the entries of the Wigner matrix have vanishing third moment. We also obtain similar results for the edge of the spectrum (but with a different scaling).

Our arguments are rather different from those of Erdos, Yau, and Yin, and thus provide an alternate approach to establishing eigenvalue concentration. The main tool is the Lindeberg exchange strategy, which is also used to prove the Four Moment Theorem (although we do not directly invoke the Four Moment Theorem in our analysis). The main novelty is that this exchange strategy is now used to establish large deviation estimates (i.e. exponentially small tail probabilities) rather than universality of the limiting distribution. Roughly speaking, the basic point is as follows. The Lindeberg exchange strategy seeks to compare a function of many independent random variables with the same function of a different set of random variables (which match moments with the original set of variables to some order, such as to second or fourth order) by exchanging the random variables one at a time. Typically, one tries to upper bound expressions such as

for various smooth test functions , by performing a Taylor expansion in the variable being swapped and taking advantage of the matching moment hypotheses. In previous implementations of this strategy, was a bounded test function, which allowed one to get control of the bulk of the distribution of , and in particular in controlling probabilities such as

for various thresholds and , but did not give good control on the tail as the error terms tended to be polynomially decaying in rather than exponentially decaying. However, it turns out that one can modify the exchange strategy to deal with moments such as

for various moderately large (e.g. of size comparable to ), obtaining results such as

after performing all the relevant exchanges. As such, one can then use large deviation estimates on to deduce large deviation estimates on .

In this paper we also take advantage of a simplification, first noted by Erdos, Yau, and Yin, that Four Moment Theorems become somewhat easier to prove if one works with resolvents (and the closely related Stieltjes transform ) rather than with individual eigenvalues, as the Taylor expansion of resolvents are very simple (essentially being a Neumann series). The relationship between the Stieltjes transform and the location of individual eigenvalues can be seen by taking advantage of the identity

for any energy level , which can be verified from elementary calculus. (In practice, we would truncate near zero and near infinity to avoid some divergences, but this is a minor technicality.) As such, a concentration result for the Stieltjes transform can be used to establish an analogous concentration result for the eigenvalue counting functions , which in turn can be used to deduce concentration results for individual eigenvalues by some basic combinatorial manipulations.

We now pursue the second approach more thoroughly. The main difficulty here is to figure out how to ensure flattening of the random walk, as it is then an easy matter to use quasirandomness to show that the random walk becomes mixing soon after it becomes flat. In the case of Selberg’s theorem, we achieved this through an explicit formula for the heat kernel on the hyperbolic plane (which is a proxy for the random walk). However, in most situations such an explicit formula is not available, and one must develop some other tool for forcing flattening, and specifically an estimate of the form

for some , where is the uniform probability measure on the generating set .

In 2006, Bourgain and Gamburd introduced a general method for achieving this goal. The intuition here is that the main obstruction that prevents a random walk from spreading out to become flat over the entire group is if the random walk gets trapped in some proper subgroup of (or perhaps in some coset of such a subgroup), so that remains large for some moderately large . Note that

since , , and is symmetric. By iterating this observation, we seethat if is too large (e.g. of size for some comparable to ), then it is not possible for the random walk to converge to the uniform distribution in time , and so expansion does not occur.

A potentially more general obstruction of this type would be if the random walk gets trapped in (a coset of) an approximate group . Recall that a -approximate group is a subset of a group which is symmetric, contains the identity, and is such that can be covered by at most left-translates (or equivalently, right-translates) of . Such approximate groups were studied extensively in last quarter’s course. A similar argument to the one given previously shows (roughly speaking) that expansion cannot occur if is too large for some coset of an approximate group.

It turns out that this latter observation has a converse: if a measure does not concentrate in cosets of approximate groups, then some flattening occurs. More precisely, one has the following combinatorial lemma:

Lemma 1 (Weighted Balog-Szemerédi-Gowers lemma) Let be a group, let be a finitely supported probability measure on which is symmetric (thus for all ), and let . Then one of the following statements hold:

• (i) (Flattening) One has .
• (ii) (Concentration in an approximate group) There exists an -approximate group in with and an element such that .

This lemma is a variant of the more well-known Balog-Szemerédi-Gowers lemma in additive combinatorics due to Gowers (which roughly speaking corresponds to the case when is the uniform distribution on some set ), which in turn is a polynomially quantitative version of an earlier lemma of Balog and Szemerédi. We will prove it below the fold.

The lemma is particularly useful when the group in question enjoys a product theorem, which roughly speaking says that the only medium-sized approximate subgroups of are trapped inside genuine proper subgroups of (or, contrapositively, medium-sized sets that generate the entire group cannot be approximate groups). The fact that some finite groups (and specifically, the bounded rank finite simple groups of Lie type) enjoy product theorems is a non-trivial fact, and will be discussed in later notes. For now, we simply observe that the presence of the product theorem, together with quasirandomness and a non-concentration hypothesis, can be used to demonstrate expansion:

Theorem 2 (Bourgain-Gamburd expansion machine) Suppose that is a finite group, that is a symmetric set of generators, and that there are constants with the following properties.

1. (Quasirandomness). The smallest dimension of a nontrivial representation of is at least ;
2. (Product theorem). For all there is some such that the following is true. If is a -approximate subgroup with then generates a proper subgroup of ;
3. (Non-concentration estimate). There is some even number such that

   

   where the supremum is over all proper subgroups .

Then is a two-sided -expander for some depending only on , and the function (and this constant is in principle computable in terms of these constants).

This criterion for expansion is implicitly contained in this paper of Bourgain and Gamburd, who used it to establish the expansion of various Cayley graphs in for prime . This criterion has since been applied (or modified) to obtain expansion results in many other groups, as will be discussed in later notes.

— 1. The Balog-Szemerédi-Gowers lemma —

The Balog-Szemerédi-Gowers lemma (Lemma 1) is ostensibly a statement about group structure, but the main tool in its proof is a remarkable graph-theoretic lemma (also known as the Balog-Szemerédi-Gowers lemma) that allows one to upgrade a “statistical” structure (a structure which is only valid a small fraction of the time, say 1% of the time) to a “complete” structure (one which is valid 100% of the time), by shrinking the size of the structure slightly (and in particular, with losses of polynomial type, as opposed to exponential or worse). This is in contrast to other structure-improving results (such as Ramsey’s theorem, Szemerédi’s theorem, or Freiman’s theorem), which are qualitatively similar in spirit, but have much worse quantitative bounds (though there is some hope in the case of Freiman’s theorem to only lose polynomial bounds with some improvement of existing arguments).

As we shall see later, the property of being large is a statistical assertion about (it asserts that collides with itself somewhat often), whereas approximate groups represent a more complete sort of structure (all products of are trapped in a small set, whereas only many of the products in are so constrained). The graph-theoretic Balog-Szemerédi lemma is the key to moving from the former type of structure to the latter with only polynomial losses.

We need some notation. Define a bipartite graph to be a graph whose vertex set is partitioned into two non-empty sets , and the edge set consists only of edges between and . If a finite bipartite graph is dense in the sense that its edge density is large, then for many vertices and , and are connected by a path of length one (i.e. an edge). It is thus intuitive that many pairs of vertices and will be connected by many paths of length two. Perhaps surprisingly, one can upgrade “many pairs” here to “almost all pairs”, provided that one is willing to shrink the set slightly. More precisely, one has

Lemma 3 (Balog-Szemerédi-Gowers lemma: paths of length two) Let be a finite bipartite graph with . Let . Then there exists a subset of with such that at least of the pairs are such that are connected by at least paths of length two (i.e. there exists at least vertices such that both lie in ).

Remark 1 It is not possible to remove the entirely from this lemma; see Exercise 6.4.2 of my book with Van Vu for a counterexample (involving Hamming balls).

Proof: The idea here is to use a probabilistic construction, picking to be a neighbourhood of a randomly selected element of . The rationale here is that if a pair of vertices in are not connected by many paths of length two, then they are unlikely to lie in the same neighbourhood, and so are unlikely to “wreck” the construction.

We turn to the details. Let be chosen uniformly at random, and let be the neighbourhood of . Observe that the expected size of is

By Cauchy-Schwarz, we conclude in particular that

Now, call a pair bad if it is connected by fewer than paths of length two, and let be the number of bad pairs in . We consider the quantity . Observe that if is a bad pair in , then there are at most values of for which and will both lie in , and so this bad pair contributes at most to the expectation. Since there are at most bad pairs, we conclude that

Combining this with (2), we see that

In particular, there exists a choice of for which the expression on the left-hand side is non-negative. This implies that

and

and the claim follows.

Given that almost all pairs in are joined by many paths of length two, it is then plausible that almost all pairs , are joined by many paths of length three, for some large subset of . Remarkably, one can now upgrade “almost all” pairs here to all pairs:

Lemma 4 (Balog-Szemerédi-Gowers lemma: paths of length three) Let be a finite bipartite graph with . Then there exists subsets of respectively with and , such that for every and , and are joined by paths of length three.

Remark 2 A lemma similar to this was first established Balog and Szemerédi, as a consequence of the Szemereédi regularity lemma. However, as a consequence of using that lemma, the polynomial bounds in the above lemma had to be replaced by much worse bounds (of tower-exponential type in ), which turns out to be far too weak for the purposes of establishing expansion.

Proof: The idea is to first prune a few “unpopular” vertices from and and then apply the preceding lemma.

Let be the vertices in of degree at least , and let be the edges connecting and . Note that the vertices in are connected to a total of at most edges, and so . Since , we conclude in particular that .

Let be a sufficiently small quantity (depending on ) to be chosen later. Applying Lemma 3, one can find a subset of of cardinality such that at most of the pairs are bad in the sense that they are connected by paths of length two.

Let be those vertices in for which there are at most elements of for which is bad. By Markov’s inequality, consists of all but at most elements of .

Let be the edges connecting with . Since each vertex in has degree at least , one has

We may thus find a subset of of cardinality such that each is adjacent to elements of .

Now let and . We know that is adjacent to elements of , and that at most of these elements are such that is bad. If we choose to be a sufficiently small multiple of , we conclude that there are elements which are adjacent to and for which is not bad. One thus has paths of length three connecting to , and the claim follows.

The exponents in here can be improved slightly, but we will not attempt to obtain the optimal numerology here.

Remark 3 The above results are analogous to a phenomenon in additive combinatorics, namely that a “1%-structured” set (such as a small density subset of a group) can often be upgraded to a “99%-structured” set (such as the complement of a small density subset of a group) by applying a single “convolution” or “sumset” operation, and then upgraded further to a “100%-structured” set (such as a genuine group) by applying a further convolution or sumset operation. (This is basically why, for instance, it is known that almost all even natural numbers are the sum of two primes, and all but finitely many odd natural numbers are the sum of three primes; but it is not known whether all but finitely many even natural numbers are the sum of two primes.)

Exercise 1 (Weighted Balog-Szemerédi-Gowers theorem) Let and be probability spaces, and let have measure for some .

• (i) Show that for any , there exists a subset of of measure such that

  

• (ii) Show that there exists subsets of of measure and such that

  

  for all and .

Exercise 2 (99% Balog-Szemerédi theorem) Let be a finite bipartite graph with .

• (i) Show that there exists a subset of of size such that for every , and are connected by at least paths of length . (Hint: select to be those vertices in that are connected to “almost all” the vertices in .)
• (ii) Show that there also exists a subset of of size such that for every and , and are connected by at least paths of length .

We now apply the graph-theoretic lemma to the group context. The main idea here is to show that various sets (e.g. product sets ) are small by showing that they are in the high-multiplicity region of some convolution (e.g. ), or equivalently that elements of such sets have many representations as a product with . One can then use Markov’s inequality and the trivial identity to get usable size bounds on such sets.

Corollary 5 (Balog-Szemerédi lemma, product set form) let be finite non-empty subsets of a group , and suppose that

for some . (This hypothesis should be compared with the upper bound

arising from Young’s inequality.) Then there exists subsets of respectively with and with and .

The quantity (or equivalently, the number of solutions to the equation with and ) is also known as the multiplicative energy of and , and is sometimes denoted in the literature.

Proof: By hypothesis, we have

Since

we conclude that

Since, by Cauchy-Schwarz (or Young’s inequality), we have , we conclude that there is a set with such that

for all .

By slight abuse of notation (arising from the fact that , are not necessarily disjoint, and that is a set of ordered pairs rather than unordered pairs), we can view the triplet as a bipartite graph. Applying Lemma 4, we can find subsets of respectively with and such that for all and , one can find elements such that . In particular, we see that

Observe that . Using the identity , we note that triples for are precisely those triples with . Thus the left-hand side of (3) is equal to , where

But since

we see from Markov’s inequality that there are at most possible values for , which gives the bound .

The second bound can be proven similarly to the first (noting that any are connected by paths of length six), but can also from the former bound as follows. Observe that any element has at least representations of the form with , and hence , thus

on . On the other hand, the left-hand side has an norm of , and the bound then follows from Markov’s inequality.

Exercise 3 In the converse direction, show that if are non-empty finite subsets of with , then .

Exercise 4 If are three non-empty finite subsets of , establish the Ruzsa triangle inequality . (Hint: mimic the final part of the proof of Corollary 5.)

We now give a variant of this corollary involving approximate groups.

Lemma 6 (Balog-Szemerédi lemma, approximate group form) Let be a finite symmetric subset of a group , and suppose that

for some . Then there exists a -approximate group with such that for some .

Proof: By Corollary 5, we may find a subset with such that

By Exercise 3, this implies that

Observe that the left-hand side is equal to

We conclude that

On the other hand, we have

As a consequence, we see that if we set

for some sufficiently large absolute constant , then

and thus (for large enough)

Since , we conclude that

Also, is clearly symmetric and contains the origin.

Now let us consider an element of the product . By construction of , we can write each as a product with in at least ways. Doing so for each gives rise to a factorisation

where ; as the uniquely determine the (for fixed ), we conclude that each element of has at least such factorisations. But by (4), there are at most such tuples , and so there are at most possible values for , thus

In particular,

By the Ruzsa covering lemma (see the exercise below), this implies that is covered by left-translates of , and so is a -approximate group. Finally, from (5) one has

and thus by Exercise 3

In particular, since the support of has size , one has

for some , or equivalently that

Increasing to and taking inverses, we conclude that , and the claim follows.

Exercise 5 (Ruzsa covering lemma) Let be finite non-empty subsets of a group . Show that can be covered by at most left-translates of . (Hint: consider a maximal disjoint collection of translates of with .)

Exercise 6 (Converse to Balog-Szemerédi-Gowers) Let be a finite symmetric subset of a group , and suppose there exists a -approximate group with such that for some . Show that

Exercise 7 Let be finite non-empty subsets of a group , and suppose that . Show that there exists a -approximate group with and elements such that and .

Finally, we can prove Lemma 1. Fix . We may assume that

and we need to use this to locate an -approximate group in with and an element such that .

Let us write . Intuitively, represents the “width” of the probability meaure , as can be seen by considering the model example where is a symmetric set of cardinality (i.e. is the uniform probability measure on ). If we were actually in this model case, we could apply Lemma 6 immediately and be done. Of course, in general, need not be a uniform measure on a set of size . However, it turns out that one can use (6) to conclude that the “bulk” of is basically of this form.

More precisely, let us split , where

Observe that

and so by Young’s inequality

In a similar vein, we have

and thus by Young’s inequality (and the normalisation )

Finally, from (6) one has

Subtracting using the triangle inequality (ignoring some slight double-counting), we conclude that

If we then set , we conclude in particular that

On the other hand, from Markov’s inequality one has . Applying Lemma 6, we conclude the existence of a -approximate group with such that for some , which by definition of implies that , and the claim follows.

— 2. The Bourgain-Gamburd expansion machine —

We can now prove Theorem 2. We can assume that is sufficiently large depending on the parameters , since the claim is trivial for bounded (note that as generates , the Cayley graph will be an -expander for some ). Henceforth we allow all implied constants in the asymptotic notation to depend on .

To show expansion, it suffices from the quasirandomness hypothesis (and Proposition 4 from the preceding notes), it will suffice to show that

for some .

From Young’s inequality, is decreasing in , and is initially equal to when . We need to “flatten” the norm of as increases. We first use the non-concentration hypothesis to obtain an initial amount of flattening:

Proposition 7 For any , one has

Furthermore, we have

for all proper subgroups of and all .

Proof: By the non-concentration hypothesis, we can find such that

for all proper subgroups of . If we write as , we see that

for all . By symmetry, , and thus

If , then we may write as the convolution of a probability measure and . From this, we see that

for all , giving the claim (9). Specialising this to the case when is the trivial group, one has

Since we also have

the claim (8) then follows from Hölder’s inequality.

Now we obtain additional flattening using the product theorem hypothesis:

Lemma 8 (Flattening lemma) Suppose is such that

Then one has

for some depending only on and .

Proof: Suppose the claim fails for some to be chosen later, thus

Applying Lemma 1, we may thus find a -approximate group with

and such that

Since by (9), we see that

Meanwhile, from (10) one has

Applying the product hypothesis (assuming sufficiently small depending on and ), we conclude that generates a proper subgroup of , and thus

But this contradicts (9) (again if is sufficiently small).

Iterating the above lemma times we obtain (7) for some , as desired.

Remark 4 Roughly speaking, the three hypotheses in Theorem 2 govern three separate stages of the life cycle of the random walk and its distributions . In the early stage , the non-concentration hypotheses creates some initial spreading of this random walk, in particular ensuring that the walk “escapes” from cosets of proper subgroups. In the middle stage , the product theorem steadily flattens the distribution of the random walk, until it is very roughly comparable to the uniform distribution. Finally, in the late stage , the quasirandomness property can smooth out the random walk almost completely to obtain the mixing necessary for expansion.

where is the distance to the origin in the unit circle (or equivalently, the distance to the nearest integer, after lifting up to ). These sets play an important role in additive combinatorics and in additive number theory. For instance, they arise naturally when applying the circle method, because Bohr sets describe the oscillation of exponential phases such as .

Observe that Bohr sets enjoy the doubling property

thus doubling the Bohr set doubles both the length parameter and the radius parameter . As such, these Bohr sets resemble two-dimensional balls (or boxes). Indeed, one can view as the preimage of the two-dimensional box under the homomorphism .

Another class of finite set with two-dimensional behaviour is the class of (rank two) generalised arithmetic progressions

with and Indeed, we have

and so we see, as with the Bohr set, that doubling the generalised arithmetic progressions doubles the two defining parameters of that progression.

More generally, there is an analogy between rank Bohr sets

and the rank generalised arithmetic progressions

One of the aims of additive combinatorics is to formalise analogies such as the one given above. By using some arguments from the geometry of numbers, for instance, one can show that for any rank Bohr set , there is a rank generalised arithmetic progression for which one has the containments

for some explicit depending only on (in fact one can take ); this is (a slight modification of) Lemma 4.22 of my book with Van Vu.

In the special case when , one can make a significantly more detailed description of the link between rank one Bohr sets and rank two generalised arithmetic progressions, by using the classical theory of continued fractions, which among other things gives a fairly precise formula for the generators and lengths of the generalised arithmetic progression associated to a rank one Bohr set . While this connection is already implicit in the continued fraction literature (for instance, in the classic text of Hardy and Wright), I thought it would be a good exercise to work it out explicitly and write it up, which I will do below the fold.

It is unfortunate that the theory of continued fractions is restricted to the rank one setting (it relies very heavily on the total ordering of one-dimensional sets such as or ). A higher rank version of the theory could potentially help with questions such as the Littlewood conjecture, which remains open despite a substantial amount of effort and partial progress on the problem. At the end of this post I discuss how one can use the rank one theory to rephrase the Littlewood conjecture as a conjecture about a doubly indexed family of rank four progressions, which can be used to heuristically justify why this conjecture should be true, but does not otherwise seem to shed much light on the problem.

— 1. Continued fractions —

One can present the theory of continued fractions in a number of ways. The classical approach is simply to work out the algebraic identities relating the various partial fractions of the continued fraction expansion. A more modern approach is to rewrite everything in terms of the action of the special linear group , either on the Euclidean plane , the projective line , or the hyperbolic plane . Here, I will split the difference and adopt a geometric perspective, interpreting continued fractions in terms of the geometry of a ray and a lattice in a plane; thus the continued fraction itself will not be the fundamental object of study, but only emerge as a derived quantity from the underlying geometry.

More specifically, we consider the lattice in . We have the standard basis for this lattice. More generally, any two lattice elements with equal to forms an (oriented) basis, thanks to Cramer’s rule. (One can of course identify the pair with an element of , and indeed this would be an appropriate and modern way of thinking about these pairs; but for some mostly idiosyncratic reasons I will eschew this sort of language in this presentation.)

Given a basis for , we can define the associated cone . Note that if is a basis, then so are and ; we will call these pairs the upward shift and rightward shift of respectively. Furthermore, the cones of the two shifted bases essentially partition the cone of the original basis. More precisely, we have

and consists of a single ray with rational slope.

Now let be a positive real number, which we will take to be irrational to avoid some minor notational issues involving termination of the continued fraction process. The ray then lies in the cone , and avoids all the lattice points in except for the origin. By the preceding discussion, if is a basis of whose cone contains , then exactly one of the two shifted bases , has their cone containing .

We can then iterate this process, starting with the basis and continually narrowing the cone of the basis around to create a sequence of bases for , with , and equal to either the rightward shift or upward shift of , and the cones decreasing to . We will call the the semiconvergent basis sequence of .

We can write the semiconvergent basis sequence in coordinates as and , then the are monotone non-decreasing sequences of natural numbers that increase to infinity. The slopes of thus converge to from below and above respectively, and are known as the best lower rational approximants and best upper rational approximants of , for reasons that we shall explain shortly.

We can track the dynamics of the subconvergent basis sequence by using and as a basis for , thus measuring the extent to which and fall below and above the ray respectively. Indeed, if we write

and

then the residual errors are positive reals, which initially take the values

and evolve according to the recursion

when (rightward shift case), or

when (upward shift case). (Note that the border case does not occur when is irrational.) In particular we see that the and decrease monotonically to zero.

We may now justify the terminology of best rational approximation:

Proposition 1 (Best rational approximation) Let , and let be the last basis in the semiconvergent basis sequence for which . Then for any natural numbers and , one has

and

or

and

Proof: As form a basis for , we have

for some integers ; in particular,

This already gives the claim when and , or when and . The case cannot occur since this would make , so we are left with the case . But in this case, we have , and hence by construction, again a contradiction.

We can also track the dynamics of the semiconvergent basis sequence by introducing the relative slope of with respect to the cone , defined as the unique positive real number such that

As is irrational, the are all irrational. It is not difficult to see that the are defined by the following recursion: , and one has when , and when . Also, the former case arises when is the upward shift of , and the latter case arises when we have a rightward shift instead.

The semiconvergent basis sequence can be viewed as a sequence of upward shifts for some non-negative integer , followed by rightward shifts for some positive integer , then upward shifts for some positive integer , and so forth. By tracking what happens to the relative slope and its reciprocal , we conclude that

for some ,

for some , and so forth, leading to the familiar continued fraction expansion

We recursively define the sequence of lattice vectors by the formulae

and

for , thus

etc. We may then describe the bases in terms of the as follows: if

for some and , then one has

if is even, and

if is odd. In particular, the sequence of bases contains the sequence

as a subsequence, representing the places is the sequence in which one changes from a rightward shift to an upward shift or vice versa. In particular, if , then the slopes (known as successive convergents to ) converge to from below when is even, and from above when is odd. As bases have determinant , we also see that

By construction, we have the recurrences

and

with initial conditions and , thus

etc. From an induction we see that the are a monotonically increasing sequence of natural numbers. Indeed, they must increase at least as fast as the Fibonacci sequence ; from an easy induction we see that , for all . In particular, the must grow at least exponentially fast in .

As with the basis sequence , we can express the in the basis:

or equivalently

Then the are positive reals, with , , and

By construction, we see that the are monotone decreasing in for ; indeed, we see that are essentially the output of the Euclidean algorithm applied to the real numbers . From (1) we see that

Since and for , we see in particular that

In particular we have a fairly precise description of how well the convergents approximate :

In particular, we have

As the are best rational approximants, we thus also have

whenever .

Remark 1 From the above inequalities one can easily establish the Dirichlet approximation theorem and the (one-dimensional) Kronecker approximation theorem.

— 2. Bohr sets —

Now we compare rank one Bohr sets to rank two progressions . It turns out that the best progressions to use here come from using the denominators of a semiconvergent basis as the generators. Indeed, if

then we have

and

Thus we have

whenever

In the converse direction, suppose that , thus and one has for some integer . As is a basis for , we have

for some integers . Taking wedge products, we see that

We write for some , and similarly write for , to conclude that

and similarly

We conclude that

whenever

It is now a routine matter to optimise the parameters to obtain a good fit. Suppose for instance that

for some . If we then set

then we have

Also, since is a basis, we have , and thus

In particular,

We thus conclude from the preceding discussion that

On the other hand, we have

and similarly

and thus by the preceding discussion

and similarly

Thus, in this case, the rank one Bohr set is morally the same object (up to constants) as the rank two progression .

Things are a bit more complicated in the regime

For sake of discussion, let us suppose that is an upward shift of , so that and . From (5) we see that and hence . Also, we have , so in particular, . If we now set

then we have

and

and so

Conversely, we have

while

and so once again we have

and

Thus again we see that in this case is morally the same object as .

Note that as shrinks, the lengths shrink also, though every so often there is a discontinuity when is incremented, thus shearing the generators as well as the lengths . (Technically, there is also a discontinuity when one passes from (4) to (5), but this is an artificial discontinuity and can be eliminated by performing a smoother interpolation between these two regimes which we omit.) Among other things, the above relationships give a rough description of the cardinality of the Bohr set ; one has

and thus (by unifying all the cases) one has

whenever

It is a shame that there is no similarly precise formula known for the size of a rank two Bohr set , as this would likely lead (among other things) to a resolution of the Littlewood conjecture.

— 3. A reformulation of the Littlewood conjecture —

The Littlewood conjecture asserts that for any two real numbers , one has

The conjecture remains open, although there are some highly non-trivial partial results, such as the result of Einsiedler, Katok, and Lindenstrauss that the set of pairs for which the Littlewood conjecture fails is so small that it has Hausdorff dimension zero.

If a pair was a counterexample to the Littlewood conjecture, then

for all positive integers . In particular, one has

for all (in other words, is badly approximable by rationals, and is in particular irrational). Applying (2), we conclude that for all , or equivalently that the coefficients of the continued fraction expansion of are bounded. (Conversely, using (2) and (3) we see that if is irrational with bounded continued fraction coefficients, then it is badly approximable by rationals.) Similarly for . This is already enough to establish without much additional work that the Littlewood conjecture is true for almost all (in the sense of Lebesgue measure), although it does not come close to the stronger result of Einseidler, Katok, and Lindenstrauss mentioned above (it is known that the set of badly approximable numbers has Hausdorff dimension one).

One can reformulate the Littlewood conjecture in terms of the denominators of the convergents to , and the denominators of the convergents to :

Proposition 2 Let be two real numbers. Then the following are equivalent:

• (i) form a counterexample to the Littlewood conjecture.
• (ii) are badly approximable, and there is an such that the rank four progressions

  

  

  are proper for all .

Recall that a progression is said to be proper if all of the elements , of the progression are distinct.

Proof: Let us first show that (i) implies (ii). Let be a counterexample to the Littlewood conjecture, thus

for all non-zero integers . We have already seen that are badly approximable (so in particular and . Now let be a sufficiently small number, and suppose that the progression (6) is not proper for some . Then there is a non-trivial solution to the equation

for some integers , not all zero, with and . Let be the integer in (8), then is non-zero and . Also, from (2) we see that

and similarly

and thus

Setting small enough, we contradict (7).

Now we show that (ii) implies (i). Let be as in (ii). Suppose for contradiction that (i) fails, thus we can find a sequence of integers going off to infinity such that

where denotes a quantity that goes to zero as (or ) goes to infinity. In particular, one can find (depending on ) such that

(Indeed, one can pick to be and to be .) As the increase in a lacunary fashion (thanks to the bad approximability of ), we can find such that ; similarly we may find such that , thus

Thus lies in the Bohr sets

and

Applying the analysis of the preceding section, we conclude that lies in the rank two progressions

and

and is non-zero, and so the rank four progression

is not proper. But this contradicts (ii) for large enough.

We can now give a heuristic explanation as to why there should not be any counterexample to the Littlewood conjecture. Such a counterexample can be completely described by the continued fraction coefficients for , and the continued fraction coefficients for . These sequences need to be bounded by the bad approximability property; let us suppose that they are both bounded by some bound . Thus, for any , the number of choices for and grows at most exponentially in (it is about ). Of course, these coefficients then determine the denominators and by the recurrences given previously.

On the other hand, by the above proposition, there needs to be an such that for each and , the progression (6) is proper. We can heuristically gauge how likely this occurs by the following numerology. For fixed (and making the non-degeneracy assumptions that ), there are about different ways to form a sum of the form

with and . These sums have magnitude about . Thus, we expect that the probability that all non-trivial sums avoid zero (which is basically what is needed for (6) to be proper) to be at most for some depending only on .

On the other hand, the number of eligible pairs for which the progression (6) is non-degenerate enough to have a chance of being proper grows quadratically in . Due to the exponential growth of and , it is reasonable to suppose that the properness of each of the progressions (6) behave like independent (or at least very weakly coupled) events as vary (viewing and as being drawn uniformly from all sequences bounded by ). If we assume that this independence is so strong as to be like joint independence (and here is where we really make a serious leap of faith), then we therefore expect the probability that a randomly chosen choice of coefficients has all of the (6) proper should decay quadratic-exponentially in (i.e. it should be something like , particularly if we choose and to be comparable in magnitude). As this beats the entropy of for large enough, we thus expect that for any given , there should not in fact be any counterexamples to (ii) (and hence to the Littlewood conjecture) with this choice of parameters.

This heuristic derivation of the Littlewood conjecture uses a common (but highly nonrigorous) probabilistic argument, in which one argues that there are so many ways in which a candidate can fail to be a counterexample to the conjecture that no counterexample actually exists. This type of heuristic is common in number theory, for instance justifying the even Goldbach conjecture (for sufficiently large even numbers, at least). Unfortunately, we do not know how to rule out the existence of some bizarre “conspiracy” between the coefficients and that manage to align all the possible (and supposedly independent) failure events in such a way that an extremely lucky and exceptional choice of coefficients manages to evade all of these events and end up as a counterexample to the conjecture. But the best we can do with current technology is to show that such conspiracies, if they exist at all, are very rare (with the aforementioned result of Einsiedler, Katok, and Lindenstrauss being the strongest result currently known of this type).

In this post I would like to highlight a useful instance of the uncertainty principle, due to Hugh Montgomery, which is useful in analytic number theory contexts. Specifically, suppose we are given a complex-valued function on the integers. To avoid irrelevant issues at spatial infinity, we will assume that the support of this function is finite (in practice, we will only work with functions that are supported in an interval for some natural numbers ). Then we can define the Fourier transform by the formula

where . (In some literature, the sign in the exponential phase is reversed, but this will make no substantial difference to the arguments below.)

The classical uncertainty principle, in this context, asserts that if is localised in an interval of length , then must be “smeared out” at a scale of at least (and essentially constant at scales less than ). For instance, if is supported in , then we have the Plancherel identity

while from the Cauchy-Schwarz inequality we have

for each frequency , and in particular that

for any arc in the unit circle (with denoting the length of ). In particular, an interval of length significantly less than can only capture a fraction of the energy of the Fourier transform of , which is consistent with the above informal statement of the uncertainty principle.

Another manifestation of the classical uncertainty principle is the large sieve inequality. A particularly nice formulation of this inequality is due independently to Montgomery and Vaughan and Selberg: if is supported in , and are frequencies in that are -separated for some , thus for all (where denotes the distance of to the origin in ), then

The reader is encouraged to see how this inequality is consistent with the Plancherel identity (1) and the intuition that is essentially constant at scales less than . The factor can in fact be amplified a little bit to , which is essentially optimal, by using a neat dilation trick of Paul Cohen, in which one dilates to (and replaces each frequency by their roots), and then sending (cf. the tensor product trick); see this survey of Montgomery for details. But we will not need this refinement here.

In the above instances of the uncertainty principle, the concept of narrow support in physical space was formalised in the Archimedean sense, using the standard Archimedean metric on the integers (in particular, the parameter is essentially the Archimedean diameter of the support of ). However, in number theory, the Archimedean metric is not the only metric of importance on the integers; the -adic metrics play an equally important role; indeed, it is common to unify the Archimedean and -adic perspectives together into a unified adelic perspective. In the -adic world, the metric balls are no longer intervals, but are instead residue classes modulo some power of . Intersecting these balls from different -adic metrics together, we obtain residue classes with respect to various moduli (which may be either prime or composite). As such, another natural manifestation of the concept of “narrow support in physical space” is “vanishes on many residue classes modulo “. This notion of narrowness is particularly common in sieve theory, when one deals with functions supported on thin sets such as the primes, which naturally tend to avoid many residue classes (particularly if one throws away the first few primes).

In this context, the uncertainty principle is this: the more residue classes modulo that avoids, the more the Fourier transform must spread out along multiples of . To illustrate a very simple example of this principle, let us take , and suppose that is supported only on odd numbers (thus it completely avoids the residue class ). We write out the formulae for and :

If is supported on the odd numbers, then is always equal to on the support of , and so we have . Thus, whenever has a significant presence at a frequency , it also must have an equally significant presence at the frequency ; there is a spreading out across multiples of . Note that one has a similar effect if was supported instead on the even integers instead of the odd integers.

A little more generally, suppose now that avoids a single residue class modulo a prime ; for sake of argument let us say that it avoids the zero residue class , although the situation for the other residue classes is similar. For any frequency and any , one has

From basic Fourier analysis, we know that the phases sum to zero as ranges from to whenever is not a multiple of . We thus have

In particular, if is large, then one of the other has to be somewhat large as well; using the Cauchy-Schwarz inequality, we can quantify this assertion in an sense via the inequality

Let us continue this analysis a bit further. Now suppose that avoids residue classes modulo a prime , for some . (We exclude the case as it is clearly degenerates by forcing to be identically zero.) Let be the function that equals on these residue classes and zero away from these residue classes, then

Using the periodic Fourier transform, we can write

for some coefficients , thus

Some Fourier-analytic computations reveal that

and

and so after some routine algebra and the Cauchy-Schwarz inequality, we obtain a generalisation of (3):

Thus we see that the more residue classes mod we exclude, the more Fourier energy has to disperse along multiples of . It is also instructive to consider the extreme case , in which is supported on just a single residue class ; in this case, one clearly has , and so spreads its energy completely evenly along multiples of .

In 1968, Montgomery observed the following useful generalisation of the above calculation to arbitrary modulus:

Proposition 1 (Montgomery’s uncertainty principle) Let be a finitely supported function which, for each prime , avoids residue classes modulo for some . Then for each natural number , one has

where is the quantity

where is the Möbius function.

We give a proof of this proposition below the fold.

Following the “adelic” philosophy, it is natural to combine this uncertainty principle with the large sieve inequality to take simultaneous advantage of localisation both in the Archimedean sense and in the -adic senses. This leads to the following corollary:

Corollary 2 (Arithmetic large sieve inequality) Let be a function supported on an interval which, for each prime , avoids residue classes modulo for some . Let , and let be a finite set of natural numbers. Suppose that the frequencies with , , and are -separated. Then one has

where was defined in (4).

Indeed, from the large sieve inequality one has

while from Proposition 1 one has

whence the claim.

There is a great deal of flexibility in the above inequality, due to the ability to select the set , the frequencies , the omitted classes , and the separation parameter . Here is a typical application concerning the original motivation for the large sieve inequality, namely in bounding the size of sets which avoid many residue classes:

Corollary 3 (Large sieve) Let be a set of integers contained in which avoids residue classes modulo for each prime , and let . Then

where

Proof: We apply Corollary 2 with , , , , and . The key point is that the Farey sequence of fractions with and is -separated, since

whenever are distinct fractions in this sequence.

If, for instance, is the set of all primes in larger than , then one can set for all , which makes , where is the Euler totient function. It is a classical estimate that

Using this fact and optimising in , we obtain (a special case of) the Brun-Titchmarsh inequality

where is the prime counting function; a variant of the same argument gives the more general Brun-Titchmarsh inequality

for any primitive residue class , where is the number of primes less than or equal to that are congruent to . By performing a more careful optimisation using a slightly sharper version of the large sieve inequality (2) that exploits the irregular spacing of the Farey sequence, Montgomery and Vaughan were able to delete the error term in the Brun-Titchmarsh inequality, thus establishing the very nice inequality

for any natural numbers with . This is a particularly useful inequality in non-asymptotic analytic number theory (when one wishes to study number theory at explicit orders of magnitude, rather than the number theory of sufficiently large numbers), due to the absence of asymptotic notation.

I recently realised that Corollary 2 also establishes a stronger version of the “restriction theorem for the Selberg sieve” that Ben Green and I proved some years ago (indeed, one can view Corollary 2 as a “restriction theorem for the large sieve”). I’m placing the details below the fold.

— 1. Proof of uncertainty principle —

We now prove Proposition 1. As with the case when is prime, the idea is to work by duality, testing against a suitably chosen test function and using the Cauchy-Schwarz inequality.

By replacing with the modulated function , we may assume without loss of generality that . We may of course assume that is square-free, and that for all , since otherwise and the claim is vacuously true. Let us abbreviate the summation as , thus our task is to show that

For each prime dividing , let be the union of the residue classes modulo which avoid . It turns out that the optimal choice for is the function

On the one hand, we see that is equal to on the support of , and thus

On the other hand, each is mean zero and periodic of period , and thus a linear combination of phases with coprime to . Multiplying together, we conclude that

for some coefficients . Taking the inner product against , we conclude that

By Cauchy-Schwarz, we conclude that

But by the Plancherel identity we have

Note that has mean , and so

Putting everything together, we obtain (5) as required.

Remark 1 The factor of in the uncertainty principle is sharp, as can be seen by computing what happens when .

— 2. Restriction theory of the large sieve —

The hypotheses of Corollary 2 are somewhat inconvenient to work with. We can specialise Corollary 2 to a more tractable version:

Theorem 4 (Special case of large sieve inequality) Let be a function supported on an interval which, for each prime , avoids residue classes modulo for some . Let , and . Then one has

where is a set of at most primes.

Proof: Observe that for each , there is at most one fraction with such that . Indeed, if there were two such fractions , then we would have

by the triangle inequality. On the other hand, the left-hand side is at least , contradicting the definition of .

By selecting at most one prime for each of the pairs with , we may thus find a natural number that is the product of at most primes, and with the property that

whenever , , and . From this (and (2)) we see that

whenever and with , with either or . Applying Corollary 2, we conclude that

On the other hand, from the multiplicative (and non-negative) nature of we have

Writing as the primes dividing , we see that

The claim follows.

Suppose that for all primes and some fixed . Then from Mertens’ theorem, we have

Also, one has the standard sieve theory bound

where is the singular series

This bound can be established by a variety of techniques (e.g. by estimating the Dirichlet series for small values of ), and can for instance be found in Lemma 4.1 of this text of Halberstam and Richert. Putting this together, we conclude that

Setting , we can simplify this a bit to

Note the very slow growth in . It is not difficult to use this bound to obtain the variant

for any and any -separated . Averaging, we obtain a restriction theorem

which is essentially Proposition 4.2 of my paper with Ben Green (but with the Selberg sieve replaced by the large sieve). As such, it can be used to deduce many of the other results in that paper. For instance, one has the following strengthening of Theorem 1.1 in that paper: if is a subset of that avoids residue classes mod for each and some , then

for all . If for some , and is sufficiently large depending on , one can then show that contains arithmetic progressions of length three by repeating the arguments in Section 6 of that paper; among other things, this reproves our result that there are infinitely many progressions of length three among the Chen primes (which arises from the two-dimensional case of the above assertion).

Theorem 1 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of . Then

Remark 1 The estimate is sharp, as can be seen by considering the case when is a unit circle, and are arcs; similarly if is any compact connected group that projects onto the circle. The connectedness hypothesis is essential, as can be seen by considering what happens if and are a non-trivial open subgroup of . For locally compact connected groups which are unimodular but not compact, there is an analogous statement, but with now a Haar measure instead of a Haar probability measure, and the right-hand side replaced simply by . The case when is a torus is due to Macbeath, and the case when is a circle is due to Raikov. The theorem is closely related to the Cauchy-Davenport inequality; indeed, it is not difficult to use that inequality to establish the circle case, and the circle case can be used to deduce the torus case by considering increasingly dense circle subgroups of the torus (alternatively, one can also use Kneser’s theorem).

By inner regularity, the hypothesis that are compact can be replaced with Borel measurability, so long as one then adds the additional hypothesis that is also Borel measurable.

A short proof of Kemperman’s theorem was given by Ruzsa. In this post I wanted to record how this argument can be used to establish the following more “robust” version of Kemperman’s theorem, which not only lower bounds , but gives many elements of some multiplicity:

Theorem 2 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of . Then for any , one has

Indeed, Theorem 1 can be deduced from Theorem 2 by dividing (1) by and then taking limits as . The bound in (1) is sharp, as can again be seen by considering the case when are arcs in a circle. The analogous claim for cyclic groups for prime order was established by Pollard, and for general abelian groups by Green and Ruzsa.

Let us now prove Theorem 2. It uses a submodularity argument related to one discussed in this previous post. We fix and with , and define the quantity

for any compact set . Our task is to establish that whenever .

We first verify the extreme cases. If , then , and so in this case. At the other extreme, if , then from the inclusion-exclusion principle we see that , and so again in this case (since ).

To handle the intermediate regime when lies between and , we rely on the submodularity inequality

for arbitrary compact . This inequality comes from the obvious pointwise identity

whence

and thus (noting that the quantities on the left are closer to each other than the quantities on the right)

at which point (2) follows by integrating over and then using the inclusion-exclusion principle.

Now introduce the function

for . From the preceding discussion vanishes at the endpoints ; our task is to show that is non-negative in the interior region . Suppose for contradiction that this was not the case. It is easy to see that is continuous (indeed, it is even Lipschitz continuous), so there must be at which is a local minimum and not locally constant. In particular, . But for any with , we have the translation-invariance

for any , and hence by (2)

Note that depends continuously on , equals when is the identity, and has an average value of . As is connected, we thus see from the intermediate value theorem that for any , we can find such that , and thus by inclusion-exclusion . By definition of , we thus have

Taking infima in (and noting that the hypotheses on are independent of ) we conclude that

for all . As is a local minimum and is arbitrarily small, this implies that is locally constant, a contradiction. This establishes Theorem 2.

We observe the following corollary:

Corollary 3 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of , and let . Then one has the pointwise estimate

if , and

if .

Once again, the bounds are completely sharp, as can be seen by computing when are arcs of a circle. For quasirandom , one can do much better than these bounds, as discussed in this recent blog post; thus, the abelian case is morally the worst case here, although it seems difficult to convert this intuition into a rigorous reduction.

Proof: By cyclic permutation we may take . For any

we can bound

where we used Theorem 2 to obtain the third line. Optimising in , we obtain the claim.

Freiman’s lemma.  Let A be a finite subset of a Euclidean space with .  Then A is contained in an affine subspace of dimension at most .

This can be viewed as a “cheap” version of the more well known theorem of Freiman that places sets of small doubling in a torsion-free abelian group inside a generalised arithmetic progression.  The advantage here is that the bound on the dimension is extremely explicit.

Our main result is

Theorem.  Let A be a finite subset of a simply-connected nilpotent Lie group G which is a K-approximate group (i.e. A is symmetric, contains the identity, and can be covered by up to K left translates of A.  Then A can be covered by at most left-translates of a closed connected Lie subgroup of dimension at most .

We remark that our previous paper established a similar result, in which the dimension bound was improved to , but at the cost of worsening the covering number to , and with a much more complicated proof  (91 pages instead of 8). Furthermore, the bound on is ineffective, due to the use of ultraproducts in the argument (though it is likely that some extremely lousy explicit bound could eventually be squeezed out of the argument by finitising everything).  Note that the step of the ambient nilpotent group G does not influence the final bounds in the theorem, although we do of course need this step to be finite.  A simple quotienting argument allows one to deduce a corollary of the above theorem in which the ambient group is assumed to be residually torsion-free nilpotent instead of being a simply connected nilpotent Lie group, but we omit the statement of this corollary here.

To motivate the proof of this theorem, let us first show a simple case of an argument of Gleason, which is very much in the spirit of Freiman’s lemma:

Gleason Lemma (special case).  Let be a finite symmetric subset of a Euclidean space, and let be a sequence of subspaces in this space, such that the sets are strictly increasing in i for .  Then , where .

Proof.    By hypothesis, for each , the projection of to is non-trivial, finite, and symmetric.   In particular, since the vector space is torsion-free, is strictly larger than .  Equivalently, one can find  in that does not lie in ; in particular, and is disjoint from .  As a consequence, the are disjoint and lie in 5A, whence the claim.

Note that by combining the contrapositive of this lemma with a greedy algorithm, one can show that any K-approximate group in a Euclidean space is contained in a subspace of dimension at most , which is a weak version of Freiman’s lemma.

To extend the argument to the nilpotent setting we use the following idea.  Observe that any non-trivial genuine subgroup H of a nilpotent group G will contain at least one non-trivial central element; indeed, by intersecting H with the lower central series of G, and considering the last intersection which is non-trivial, one obtains the claim.  It turns out that one can adapt this argument to approximate groups, so that any sufficiently large K-approximate subgroup A of G will contain a non-trivial element that centralises a large fraction of A.  Passing to this large fraction and quotienting out the central element, we obtain a new approximate group.    If, after a bounded number of steps, this procedure gives an approximate group of bounded size, we are basically done.  If, however, the process continues, then by using some Lie group theory, one can find a long sequence of connected Lie subgroups of G, such that the sets are strictly increasing in i.   Using some Lie group theory and the hypotheses on G, one can deduce that the group generated by is much larger than , in the sense that the latter group has infinite index in the former.   It then turns out that the Gleason argument mentioned above can be adapted to this setting.

which then gives a functional calculus, in the sense that the map is a -algebra isometric homomorphism from the algebra of bounded continuous functions from to , to the algebra of bounded linear operators on . Thus, for instance, one can define heat operators for , Schrödinger operators for , resolvents for , and (if is positive) wave operators for . These will be bounded operators (and, in the case of the Schrödinger and wave operators, unitary operators, and in the case of the heat operators with positive, they will be contractions). Among other things, this functional calculus can then be used to solve differential equations such as the heat equation

the Schrödinger equation

the wave equation

or the Helmholtz equation

The functional calculus can also be associated to a spectral measure. Indeed, for any vectors , there is a complex measure on with the property that

indeed, one can set to be the discrete measure on defined by the formula

One can also view this complex measure as a coefficient

of a projection-valued measure on , defined by setting

Finally, one can view as unitarily equivalent to a multiplication operator on , where is the real-valued function , and the intertwining map is given by

so that .

It is an important fact in analysis that many of these above assertions extend to operators on an infinite-dimensional Hilbert space , so long as one one is careful about what “self-adjoint operator” means; these facts are collectively referred to as the spectral theorem. For instance, it turns out that most of the above claims have analogues for bounded self-adjoint operators . However, in the theory of partial differential equations, one often needs to apply the spectral theorem to unbounded, densely defined linear operators , which (initially, at least), are only defined on a dense subspace of the Hilbert space . A very typical situation arises when is the square-integrable functions on some domain or manifold (which may have a boundary or be otherwise “incomplete”), and are the smooth compactly supported functions on , and is some linear differential operator. It is then of interest to obtain the spectral theorem for such operators, so that one build operators such as or to solve equations such as (1), (2), (3), (4).

In order to do this, some necessary conditions on the densely defined operator must be imposed. The most obvious is that of symmetry, which asserts that

for all . In some applications, one also wants to impose positiveness, which asserts that

for all . These hypotheses are sufficient in the case when is bounded, and in particular when is finite dimensional. However, as it turns out, for unbounded operators these conditions are not, by themselves, enough to obtain a good spectral theory. For instance, one consequence of the spectral theorem should be that the resolvents are well-defined for any strictly complex , which by duality implies that the image of should be dense in . However, this can fail if one just assumes symmetry, or symmetry and positive definiteness. A well-known example occurs when is the Hilbert space , is the space of test functions, and is the one-dimensional Laplacian . Then is symmetric and positive, but the operator does not have dense image for any complex , since

for all test functions , as can be seen from a routine integration by parts. As such, the resolvent map is not everywhere uniquely defined. There is also a lack of uniqueness for the wave, heat, and Schrödinger equations for this operator (note that there are no spatial boundary conditions specified in these equations).

Another example occurs when , , is the momentum operator . Then the resolvent can be uniquely defined for in the upper half-plane, but not in the lower half-plane, due to the obstruction

for all test functions (note that the function lies in when is in the lower half-plane). For related reasons, the translation operators have a problem with either uniqueness or existence (depending on whether is positive or negative), due to the unspecified boundary behaviour at the origin.

The key property that lets one avoid this bad behaviour is that of essential self-adjointness. Once is essentially self-adjoint, then spectral theorem becomes applicable again, leading to all the expected behaviour (e.g. existence and uniqueness for the various PDE given above).

Unfortunately, the concept of essential self-adjointness is defined rather abstractly, and is difficult to verify directly; unlike the symmetry condition (5) or the positive condition (6), it is not a “local” condition that can be easily verified just by testing on various inputs, but is instead a more “global” condition. In practice, to verify this property, one needs to invoke one of a number of a partial converses to the spectral theorem, which roughly speaking asserts that if at least one of the expected consequences of the spectral theorem is true for some symmetric densely defined operator , then is self-adjoint. Examples of “expected consequences” include:

• Existence of resolvents (or equivalently, dense image for );
• Existence of a contractive heat propagator semigroup (in the positive case);
• Existence of a unitary Schrödinger propagator group ;
• Existence of a unitary wave propagator group (in the positive case);
• Existence of a “reasonable” functional calculus.
• Unitary equivalence with a multiplication operator.

Thus, to actually verify essential self-adjointness of a differential operator, one typically has to first solve a PDE (such as the wave, Schrödinger, heat, or Helmholtz equation) by some non-spectral method (e.g. by a contraction mapping argument, or a perturbation argument based on an operator already known to be essentially self-adjoint). Once one can solve one of the PDEs, then one can apply one of the known converse spectral theorems to obtain essential self-adjointness, and then by the forward spectral theorem one can then solve all the other PDEs as well. But there is no getting out of that first step, which requires some input (typically of an ODE, PDE, or geometric nature) that is external to what abstract spectral theory can provide. For instance, if one wants to establish essential self-adjointness of the Laplace-Beltrami operator on a smooth Riemannian manifold (using as the domain space), it turns out (under reasonable regularity hypotheses) that essential self-adjointness is equivalent to geodesic completeness of the manifold, which is a global ODE condition rather than a local one: one needs geodesics to continue indefinitely in order to be able to (unitarily) solve PDEs such as the wave equation, which in turn leads to essential self-adjointness. (Note that the domains and in the previous examples were not geodesically complete.) For this reason, essential self-adjointness of a differential operator is sometimes referred to as quantum completeness (with the completeness of the associated Hamilton-Jacobi flow then being the analogous classical completeness).

In these notes, I wanted to record (mostly for my own benefit) the forward and converse spectral theorems, and to verify essential self-adjointness of the Laplace-Beltrami operator on geodesically complete manifolds. This is extremely standard analysis (covered, for instance, in the texts of Reed and Simon), but I wanted to write it down myself to make sure that I really understood this foundational material properly.

— 1. Self-adjointness and resolvents —

To begin, we study what we can abstractly say about a densely defined symmetric linear operator on a Hilbert space . To avoid some technical issues we shall assume that the Hilbert space is separable, which is the case typically encountered in applications (particularly in PDE). We will occasionally assume also that is positive, but will make this hypothesis explicit whenever we are doing so.

All convergence in Hilbert spaces will be in the strong (i.e. norm) topology unless otherwise stated. Similarly, all inner products and norms will be over unless otherwise stated.

For technical reasons, it is convenient to reduce to the case when is closed, which means that the graph is a closed subspace of . Equivalently, is closed if whenever is a sequence converging strongly to a limit , and converges to a limit in , then and . For instance, thanks to the closed graph theorem, an everywhere-defined linear operator is closed if and only if it is bounded.

Not every densely defined symmetric linear operator is closed (indeed, one could take a closed operator and restrict the domain of definition to a proper dense subspace). However, all such operators are closable, in that they have a closure:

Lemma 1 (Closure) Let be a densely defined symmetric linear operator. Then there exists a unique extension of as a closed, densely defined symmetric linear operator, such that the graph of is the closure of the graph of .

Proof: The key step is to show that the closure of the graph of remains a graph, i.e. it obeys the vertical line test. If this failed, then by linearity one could find a sequence converging to zero such that converged to a non-zero limit . Since is dense, we can find such that . But then by symmetry

On the other hand, as , . Thus is the graph of some function . It is easy to see that this is a densely defined symmetric linear operator extending , and is the unique such operator.

Exercise 1 Show that is positive if and only if is positive.

Remark 1 We caution that is not the closure or completion of with respect to the usual norm on . (Indeed, as is a dense subspace of , that completion of is simply .) However, it is the completion of with respect to the modified norm .

In PDE applications, the closure tends to be defined on a Sobolev space of functions that behave well at the boundary, and is given by a distributional derivative. Here is a simple example of this:

Exercise 2 Let be the Laplacian , defined on the dense subspace of . Show that the closure is defined on the Sobolev space , defined as the closure of under the Sobolev norm

and that the action of is given by the weak (distributional) derivative, .

Next, we define the adjoint of , which informally speaking is the maximally defined operator for which one has the relationship

for and . More formally, define to be the set of all vectors for which the map is a bounded linear functional on , which thus extends to the closure of . For such , we may apply the Riesz representation theorem for Hilbert spaces and locate a unique vector for which (7) holds. This is easily seen to define a linear operator . Furthermore, the claim that is symmetric can be reformulated as the claim that extends .

If is not symmetric, then does not extend , and need not be densely defined at all. However, it is still closed:

Exercise 3 Let be a densely defined linear operator, and let be its adjoint. Show that is the orthogonal complement in of (the closure of) . Conclude that is always closed.

If is symmetric, show that and .

Exercise 4 Construct an example of a densely defined linear operator in a separable Hilbert space for which is only defined at . (Hint: Build a dense linearly independent basis of , let be the algebraic span of that basis, and design so that the graph of is dense in .)

We caution that the adjoint of a symmetric densely defined operator need not be itself symmetric, despite extending the symmetric operator :

Exercise 5 We continue Example 2. Show that for any complex number , the functions lie in with . Deduce that is not symmetric, and not positive.

Intuitively, the problem here is that the domain of is too “small” (it stays too far away from the boundary), which makes the domain of too “large” (it contains too much stuff coming from the boundary), which ruins the integration by parts argument that gives symmetry.

Now we can define (essential) self-adjointness.

Definition 2 Let be a densely defined linear operator.

• is self-adjoint if . (Note that this implies in particular that is symmetric and closed.)
• is essentially self-adjoint if it is symmetric, and its closure is self-adjoint.

Note that this extends the usual definition of self-adjointness for bounded operators. Conversely, from the closed graph theorem we also observe the Hellinger-Toeplitz theorem: an operator that is self-adjoint and everywhere defined, is necessarily bounded.

Exercise 6 Let be a densely defined symmetric closed linear operator. Show that is self-adjoint if and only if is symmetric.

It is not immediately obvious what advantage self-adjointness gives. To see this, we consider the problem of inverting the operator for some complex number , where is densely defined, symmetric, and closed. Observe that if , then is necessarily real. In particular,

and hence by the Cauchy-Schwarz inequality

In particular, if is strictly complex (i.e. not real), then is injective. Furthermore, we see that if is such that is convergent, then by (8) is convergent also, and hence is convergent. As was assumed to be closed, we conclude that converges to a limit in , and converges to . As a consequence, we see that the space is a closed subspace of . From (8) we then see that we can define an inverse of , which we call the resolvent of with spectral parameter ; this is a bounded linear operator with norm .

Exercise 7 If is densely defined, symmetric, closed, and positive, and is a complex number with , show is well-defined on with .

Now we observe a connection between self-adjointness and the domain of the resolvent. We first need some basic properties of the resolvent:

Exercise 8 Let be densely defined, symmetric, and closed.

• (i) (Resolvent identity) If are distinct strictly complex numbers with everywhere defined, show that . (Hint: compute two different ways.)
• (ii) If is a strictly complex number with and everywhere defined, show that .

Proposition 3 Let be densely defined, symmetric, and closed, let be the adjoint, and let be strictly complex.

• (i) (Surjectivity is dual to injectivity) is everywhere defined if and only if the operator has trivial kernel.
• (ii) (Self-adjointness implies surjectivity) If is self-adjoint, then is everywhere defined.
• (iii) (Surjectivity implies self-adjointness) Conversely, if and are both everywhere defined, then is self-adjoint.

In particular, we see that we have a criterion for self-adjointness: a densely defined symmetric closed operator is self-adjoint if and only if and are both everywhere defined, or in other words if and are both surjective.

Proof: We first prove (i). If is not everywhere defined, then the closed subspace is not all of . Thus there must be a non-zero vector in the orthogonal complement of this subspace, thus for all . In particular,

and hence, by definition of , lies in . Now from (7) we have

for all ; since is dense, we have as required. The converse implication follows by reversing these steps.

Now we prove (ii). Suppose that was not everywhere defined. By (i) and self-adjointness, we conclude that for some non-zero . But this contradicts (8).

Now we prove (iii). Let ; our task is to show that . From Exercise 8(ii) and (7) one has

for all . We conclude that . Since takes values in , the claim follows.

Exercise 9 Let be densely defined, symmetric, and closed, and let be strictly complex.

• (i) If is everywhere defined, show that is everywhere defined whenever . (Hint: use Neumann series.)
• (ii) If is everywhere defined, show that is everywhere defined whenever has the same sign as .
• (iii) If is positive, and is not a non-negative real, show that is everywhere defined if and only if is self-adjoint.

As a particular corollary of the above exercise, we see that a densely defined, symmetric closed positive operator is self-adjoint if and only if is everywhere defined, or in other words if is surjective.

Exercise 10 Let be a measure space with a countably generated -algebra (so that is separable), let be a measurable function, and let be the space of all such that . Show that the multiplier operator defined by is a densely defined self-adjoint operator.

Exercise 11 Let , , and is the momentum operator . Show that is densely defined and symmetric, and is everywhere defined, but is only defined on the orthogonal complement of . In particular, is not self-adjoint.

Exercise 12 Let be densely defined and symmetric.

• (i) Show that is essentially self-adjoint if and only if and are dense in .
• (ii) If is positive, show that is essentially self-adjoint if and only if is dense in .

Exercise 13 Let and be sequences of real numbers, with the all non-zero. Define the Jacobi operator from the space of compactly supported sequences to the space of square-summable sequences by the formula

with the convention that vanishes for .

• (i) Show that is densely defined and symmetric.
• (ii) Show that is essentially self-adjoint if and only if the (unique) solution to the recurrence

  

  with (and the convention ), is not square-summable.

This exercise shows that the self-adjointness of an operator, even one as explicit as a Jacobi operator, can depend in a rather subtle and “global” fashion on the behaviour of the coefficients of that oeprator.

— 2. Self-adjointness and spectral measure —

We have seen that self-adjoint operators have everywhere-defined resolvents for all strictly complex . Now we use this fact to build spectral measures. We will need a useful tool from complex analysis, which places a one-to-one correspondence between finite non-negative measures on and certain analytic functions on the upper half-plane:

Theorem 4 (Herglotz representation theorem) Let be an analytic function from the upper half-plane to the closed upper half-plane , obeying a bound of the form for all and some . Then there exists a finite non-negative Radon measure on such that

for all . Furthermore, one has as .

We set the proof of the above theorem as an exercise below. Note in the converse direction that if is a finite non-negative Radon measure, then the function defined by (9) obeys all the hypotheses of the theorem. The Herglotz representation theorem, like the more well known Riesz representation theorem for measures, is a useful tool to construct non-negative Radon measures; later on we will also use Bochner’s theorem for a similar purpose.

Exercise 14 Let be as in the hypothesis of the Herglotz representation theorem. For each , let be the function .

• (i) Show that one has for all , where is the Poisson kernel and is the usual convolution operation. (Hint: apply Liouville’s theorem for harmonic functions to the difference between and .)
• (ii) Show that the non-negative measures have a finite mass independent of , and converge in the vague topology as to a non-negative finite measure .
• (iii) Prove the Herglotz representation theorem.

Now we return to spectral theory. Let be a densely-defined self-adjoint operator, and let . We consider the function

on the upper half-plane. We can use this function and the Herglotz representation theorem to construct spectral measures :

Exercise 15 Let , , and be as above.

• (i) Show that is analytic. (Hint: use Neumann series and Morera’s theorem.)
• (ii) Show that for all in the upper half-plane.
• (iii) Show that for all in the upper half-plane. (Hint: You will find Exercise 8 to be useful.)
• (iv) Show that for all . (Hint: first show this for , writing for some .)
• (v) Show that there is a non-negative Radon measure of total mass such that

  

  for all in either the upper or lower half-plane.

• (vi) If is positive, show that is supported in the right half-line .

We can depolarise these measures by defining

to obtain complex measures for any such that

for all in either the upper or lower half-plane. From duality we see that this uniquely defines . In particular, depends sesquilinearly on , and . Also, since each has mass , we see that the inner product can be recovered from the spectral measure :

Exercise 16 With the above notation and assumptions, establish the bound for all . In particular, for any bounded Borel-measurable function , there exists a unique bounded operator such that

Thus, for instance is the identity when , and when .

We have just created a map from the bounded Borel-measurable functions on , to the bounded operators on . Now we verify some basic properties of this map.

Exercise 17 (Bounded functional calculus) Let be a self-adjoint densely defined operator, and let be as above.

• (i) Show that the map is -linear. In particular, if is real-valued, then is self-adjoint.
• (ii) For any and any strictly complex , show that . (Hint: use the resolvent identity.)
• (iii) For any and any , show that .
• (iv) Show that the map is a -homomorphism. In particular, and commute for all .
• (v) Show that for all . Improve the to . (Hint: to get this improvement, use the identity and the tensor power trick.)
• (vi) For any Borel subset of , show that is an orthogonal projection of . Show that is a countably additive projection-valued measure, thus for any sequence of disjoint Borel , where the convergence is in the strong operator topology.
• (vii) For any bounded Borel set , show that the image of lies in .
• (viii) Show that for any and , one has .
• (ix) Let be the union of the images of for . Show that is a dense subspace of (and hence of ), and that maps to .
• (x) Show that is the space of all functions such that . Conclude in particular that maps to for all .
• (xi) If is such that for all , show that takes values in , that for all , and for all .
• (xii) Let be the space of all such that is not invertible. Show that is a closed set which is the union of the supports of the as range over . In particular, , and when is positive. Show that is the identity map.
• (xiii) Extend the bounded functional calculus from the bounded Borel measurable functions on to the bounded measurable functions on the spectrum (i.e. show that the previous statements (i)-(xi) continue to hold after is replaced by throughout).

The above collection of facts (or various subcollections thereof) is often referred to as the spectral theorem. It is stated for self-adjoint operators, but one can of course generalise the spectral theorem to essentially self-adjoint operators by applying the spectral theorem to the closure. (One has to replace the domain of by the domain of the closure , of course, when doing so.)

Exercise 18 (Spectral measure and eigenfunctions) Let the notation be as in the preceding exercise, and let . Show that the space is the range of the projection . In particular, has an eigenfunction at if and only if is non-trivial.

We have seen how the existence of resolvents gives us a bounded functional calculus (i.e. the conclusions in the above exercise). Conversely, if a symmetric densely defined closed operator has a bounded functional calculus, one can define the resolvents simply as , where . Thus we see that the existence of a bounded functional calculus is equivalent to the existence of resolvents, which by the previous discussion is equivalent to self-adjointness.

Using the bounded functional calculus, one can not only recover the resolvents , but can now also build Schrödinger propagators , and when is positive definite one can also build heat operators for and wave operators for (and also define resolvents for negative choices of the spectral parameter). We will study these operators more in the next section.

Exercise 19 (Locally bounded functional calculus) Let be a densely defined self-adjoint operator, and let be the space of Borel-measurable functions from to which are bounded on every bounded set.

• Show that for every there is a unique linear operator such that

  

  whenever .

• (i) Show that the map is a -homomorphism.
• (ii) Show that when is actually bounded (rather than merely locally bounded), then this definition of agrees with that in the preceding exercise (after restricting from to ).
• (iii) Show that , where is the identity map .
• (iv) State and prove a rigorous version of the formal assertion that

  

  and more generally

  

Now we use the spectral theorem to place self-adjoint operators in a normal form. Let us say that two operators and are unitarily equivalent if there is a unitary map with and . It is easy to see that all the constructions given above (such as the bounded functional calculus) are preserved by unitary equivalence.

If is a densely defined self-adjoint operator, define an invariant subspace to be a subspace of such that for all . We say that a vector is a cyclic vector for if the set is dense in .

Exercise 20

• (i) Show that if is an invariant subspace of , then so is , and furthermore the orthogonal projections to and commute with for every .
• (ii) Show that if is a invariant subspace of , then the restriction of to is a densely defined self-adjoint operator on with . Furthermore, one has for all .
• (iii) Show that decomposes as a direct sum , where the index set is at most countable, and each is a closed invariant subspace of (with the mutually orthogonal), such that each has a cyclic vector . (Hint: use Zorn’s lemma and the separability of .)
• (iv) If has a cyclic vector , show that is unitarily conjugate to a multiplication operator on for some non-negative Radon measure , defined on the domain .
• (v) For general , show that is unitarily conjugate to a multiplication operator on for some measure space with a countably generated -algebra and some measurable , defined on the domain .

The above exercise gives a satisfactory concrete description of a self-adjoint operator (up to unitary equivalence) as a multiplication operator on some measure space , although we caution that this equivalence is not canonical (there is some flexibility in the choice of the underlying measure space and multiplier , as well as the unitary conjugation map).

Exercise 21 Let be a self-adjoint densely defined operator.

• (i) Show that is positive if and only if .
• (ii) Show that is bounded if and only if is bounded. Furthermore, in this case we have .
• (iii) Show that is trivial if and only if is empty.

— 3. Self-adjointness and flows —

Now we relate self-adjointness to a variety of flows, beginning with the heat flow.

Exercise 22 Let be a self-adjoint positive densely defined operator, and for each , let be the heat operator .

• (i) Show that for each , is a bounded self-adjoint operator of norm at most , that the map is continuous in the strong operator topology, and such that and for all . (The latter two properties are asserting that is a one-parameter semigroup.)
• (ii) Show that for any , converges to as .
• (iii) Conversely, if is not in , show that does not converge as .

We remark that the above exercise can be viewed as a special case of the Hille-Yoshida theorem.

We now establish a converse to the above statement:

Theorem 5 Let be a separable Hilbert space, and suppose one has a family of bounded self-adjoint operators of norm at most for each , which is continuous in the strong operator topology, and such that and for all . Then there exists a unique self-adjoint positive densely defined operator such that for all .

We leave the proof of this result to the exercise below. The basic idea is to somehow use the identity

which suggests that

which should allow one to recover from the .

Exercise 23 Let the notation be as in the above theorem.

• (i) Establish the uniqueness claim. (Hint: use Exercise 22.)
• (ii) Let be the operator

  

  Show that is well-defined, bounded, positive semidefinite, and self-adjoint, with operator norm at most one, and that commutes with all the .

• (iii) Show that the spectrum of lies in , but that has no eigenvalue at .
• (iv) Show that there exists a densely defined self-adjoint operator such that is the identity on and is the identity on .
• (v) Show that is densely defined self-adjoint and positive definite, and commutes with all the .
• (vi) Show that for all and , one has and , where the derivatives are in the classical limiting Newton quotient sense (in the strong topology of ).
• (vii) Conclude the proof of Theorem 5. (Hint: show that is non-positive.)

The above exercise gives an important way to establish essential self-adjointness, namely by solving the heat equation (1):

Exercise 24 Let be a densely defined symmetric positive definite operator. Suppose that for every there exists a continuously differentiable solution to (1). Show that is essentially self-adjoint. (Hint: by investigating , establish the uniqueness of solutions to the heat equation, which allows one to define linear contractions . To establish self-adjointness of the , take an inner product of a solution to the heat equation against a time-reversed solution to the heat equation, and differentiate that inner product in time. Now apply the preceding exercises to obtain a self-adjoint extension of . To show that is the closure of , it suffices to show that is dense in with the inner product . But if is not dense in , then it has a non-trivial orthogonal complement; apply to this complement to show that also has a non-trivial orthogonal complement in , a contradiction.)

Remark 2 When applying the above criterion for essential self-adjointness, one usually cannot use the space of compactly supported smooth functions as the dense subspace, because this space is usually not preserved by the heat flow. However, in practice one can get around this by enlarging the class, for instance to the class of Schwartz functions.

Now we obtain analogous results for the Schrödinger propagators . We begin with the analogue of Exercise 22:

Exercise 25 Let be a self-adjoint densely defined operator, and for each , let be the Schrödinger operator .

• (i) Show that for each , is a unitary operator, that the map is continuous in the strong operator topology, and such that and for all .
• (ii) Show that for any , converges to as .
• (iii) Conversely, if is not in , show that does not converge as .

Now we can give the converse, known as Stone’s theorem on one-parameter unitary groups:

Theorem 6 (Stone’s theorem) Let be a separable Hilbert space, and suppose one has a family of unitary for each , which is continuous in the strong operator topology, and such that and for all . Then there exists a unique self-adjoint densely defined operator such that for all .

We outline a proof of this theorem in an exercise below, based on using the group to build spectral measure.

Exercise 26 Let the notation be as in the above theorem.

• (i) Establish the uniqueness component of Stone’s theorem.
• (ii) Show that for any , there is a non-negative Radon measure of total mass such that

  

  for all . (Hint: use Bochner’s theorem, a proof of which (at least on , which is the case of interest here) can be found for instance in these notes.)

• (iii) Show that for any , there is a unique complex measure such that

  

  for all . Show that is sesquilinear in with .

• (iv) Show that for any , there is a unique bounded operator such that

  

  for all .

• (v) Show that for all and .
• (vi) Show that the map is a *-homomorphism from to .
• (vii) Show that there is a projection-valued measure with , such that for every , the complex measure is such that

  

  for all .

• (viii) Show that there exists a self-adjoint densely defined operator whose spectral measure is .
• (ix) Conclude the proof of Stone’s theorem.

Exercise 27 Let be a densely defined symmetric operator. Suppose that for every there exists a continuously differentiable solution to (2). Show that is essentially self-adjoint.

We can now see a clear link between essential self-adjointness and completeness, at least in the case of scalar first-order differential operators:

Exercise 28 Let be a smooth manifold with a smooth measure , and let be a smooth vector field on which is divergence-free with respect to the measure . Suppose that the vector field is complete in the sense that for any , there exists a global smooth solution to the ODE with initial data . Show that the first-order differential operator is essentially self-adjoint.

Extend the above result to non-divergence-free vector fields, after replacing with .

Remark 3 The requirement of completeness is basically necessary; one can still have essential self-adjointness if there are a measure zero set of initial data for which the trajectories of are incomplete, but once a positive measure set of trajectories become incomplete, the propagators do not make sense globally, and so self-adjointness should fail.

Finally, we turn to the relationship between self-adjointness and the wave equation, which is a more complicated variant of the relationship between self-adjointness and the Schrödinger equation. More precisely, we will show the following version of Exercise 27:

Theorem 7 Let be a densely defined positive symmetric operator. Suppose for every , there exists a twice continuously differentiable (in ) solution to (3). Then is essentially self-adjoint.

(One could also obtain wave equation analogues of Exercise 25 or Theorem 6, but these are somewhat messy to state, and we will not do so here.)

We now prove this theorem. To simplify the exposition, we will assume that is strictly positive, in the sense that for all non-zero , and leave the general case as an exercise.

We introduce a new inner product on by the formula

By hypothesis, this is a Hermitian inner product on . We then define an inner product on by the formula

then this is a Hermitian inner product on . We define the energy space be the completion of with respect to this inner product; we can factor this as , where is the completion of using the inner product.

Suppose is a twice continuously differentiable solution to (3) for some . Then if we define the energy

then one easily computes using (3) that , and so

In particular, if , then , and so twice continuously differentiable solutions to (3) are unique. This allows us to define wave operators by defining . This is clearly linear, and from the energy identity we see that is an isometry, and thus extends to an isometry on the energy space . From uniqueness we also see that is a one-parameter group, i.e. a homomorphism that is continuous in the strong operator topology. In particular, the isometries are invertible and are thus unitary. By Stone’s theorem, there thus exists a densely defined self-adjoint operator on some dense subspace of such that for all .

If , then from the twice differentiability of the solution to the wave equation, we see that

From Exercise 2, we conclude that and

for all .

Now we need to pass from the self-adjointness of to the essential self-adjointness of . Suppose for contradiction that was not essentially self-adjoint. Then does not have dense image, and so there exists a non-zero such that

for all .

It will be convenient to work with a “band-limited” portion of , to get around the problem that and can leave this domain. Let be a compactly supported even smooth function of total mass one. For any , define the Littlewood-Paley projection

where is the Schwartz function . Then (by strong continuity of ) these operators map to , commutes with the entire functional calculus of , and also maps to . Also, from functional calculus one sees that

and in particular maps to as well.

Let be the reflection operator . From time reversal of the wave equation, we have , and thus commutes with . In particular, we can write

for some operators and . Since preserves , the operators preserve .

For any , we have

combining this with (12) we see that

and

for all . Also, since preserves , we see that preserves .

We can then expand as

and thus

In particular

for all . By duality (noting that is a bounded real function of ) we conclude that

But by sending and using the spectral theorem, this implies from monotone convergence that the spectral measure is zero, and thus vanishes in , a contradiction. This establishes the essential self-adjointness of .

Exercise 29 Establish Theorem 7 without the assumption that is strictly positive. (Hint: use the Cauchy-Schwarz inequality to show that strict positivity is equivalent to the absence of eigenfunctions with eigenvalue zero, and then quotient out such eigenfunctions.)

— 4. Essential self-adjointness of the Laplace-Beltrami operator —

We now discuss how one can use the above criteria to establish essential self-adjointness of a Laplace-Beltrami operator on a smooth complete Riemannian manifold , viewed as a densely defined symmetric positive operator on the dense subspace of . This result was first established by Gaffney and by Roelcke.

To do this, we have to solve a PDE – either the Helmholtz equation (4) (for some not on the positive real axis), the heat equation (1), the Schrödinger equation (25), or the wave equation (3).

The Schrödinger formalism is quite suggestive. From a semiclassical perspective, the Schrödinger equation associated to the Laplace-Beltrami operator should be viewed as a quantum version of the classical flow associated to the corresponding Hamiltonian , i.e. geodesic flow. From Exercise 28 we know that the generator of Hamiltonian flows (normalised by ) are essentially self-adjoint when they are complete (note from Liouville’s theorem that such generators are automatically divergence-free with respect to Liouville measure), which suggests that the Schrödinger operator should be also. Unfortunately, this is not a rigorous argument, and is difficult to make it so due to the nature of the time-dependent Schrödinger equation, which has infinite speed of propagation, and no dissipative properties. In practice, we therefore establish esssential self-adjointness by solving one of the other equations.

To solve any of these equations, it is difficult to solve any of them by an exact formula (unless the manifold is extremely symmetric); but one can proceed by solving them approximately, and using perturbation theory to eliminate the error. This method works as long as the error created by the approximate solution is both sufficiently small and sufficiently smooth that perturbative techniques (such as Neumann series, or the inverse function theorem) become applicable. (This type of method, for instance, is used to deduce essential self-adjointness of various perturbations of the Laplace-Beltrami operator from the essential self-adjointness of the original operator .)

For instance, suppose one is trying to solve the Helmholtz equation

for some large real , and some . For sake of concreteness let us take to be three-dimensional. If was a Euclidean space , then we have an explicit formula

for the solution; note that the exponential decay of will keep in as well. Inspired by this, we can try to solve the Helmholtz equation in curved space by proposing as an approximate solution

This is a little problematic because develops singularities after a certain point, but if one has a uniform lower bound on the injectivity radius (here we are implicitly using the hypothesis of completeness), and also uniform bounds on the curvature and its derivatives, one can truncate this approximate solution to the region where is small, and obtain an approximate solution whose error

can be made to be smaller in norm than that of if the spectral parameter is chosen large enough; we omit the details. From this, we can then solve the Helmholtz equation by Neumann series and thus establish essential self-adjointness. A similar method also works (under the same hypotheses on ) to construct approximate heat kernels, giving another way to establish essential self-adjointness of the Laplace-Beltrami operator.

What about when there is no uniform bound on the geometry? In that case, it is best to work with the wave equation formalism, because the finite speed of propagation property of that equation allows one to localise to compact portions of the manifold for which uniform bounds on the geometry are automatic. Indeed, to solve the wave equation in for some fixed period of time, one can use finite speed of propagation, combined with completeness of the manifold, to work in a compact subset of this manifold, which by suitable alteration of the metric beyond the support of the solution, one can view as a subset of a compact complete manifold. On such manifolds, we already know essential self-adjointness by the previous arguments, so we may solve the wave equation in that setting (one can also solve the wave equation using methods from microlocal analysis, if desired), and then take repeated advantage of finite speed of propagation (which can be proven rigorously by energy methods) to glue together these local solutions to obtain a global solution; we omit the details.

In all these cases, a somewhat nontrivial application of PDE theory is required. Unfortunately, this seems to be inevitable; at some point one must somehow use the hypothesis of completeness of the underlying manifold, and PDE methods are the only known way to connect that hypothesis to the dynamics of the Laplace-Beltrami operator.

The definition of quasirandomness is easy enough to state:

Definition 1 (Quasirandom groups) Let be a finite group, and let . We say that is -quasirandom if all non-trivial unitary representations of have dimension at least . (Recall a representation is trivial if is the identity for all .)

Exercise 1 Let be a finite group, and let . A unitary representation is said to be irreducible if has no -invariant subspaces other than and . Show that is -quasirandom if and only if every non-trivial irreducible representation of has dimension at least .

Remark 1 The terminology “quasirandom group” was introduced explicitly (though with slightly different notational conventions) by Gowers in 2008 in his detailed study of the concept; the name arises because dense Cayley graphs in quasirandom groups are quasirandom graphs in the sense of Chung, Graham, and Wilson, as we shall see below. This property had already been used implicitly to construct expander graphs by Sarnak and Xue in 1991, and more recently by Gamburd in 2002 and by Bourgain and Gamburd in 2008. One can of course define quasirandomness for more general locally compact groups than the finite ones, but we will only need this concept in the finite case. (A paper of Kunze and Stein from 1960, for instance, exploits the quasirandomness properties of the locally compact group to obtain mixing estimates in that group.)

Quasirandomness behaves fairly well with respect to quotients and short exact sequences:

Exercise 2 Let be a short exact sequence of finite groups .

• (i) If is -quasirandom, show that is -quasirandom also. (Equivalently: any quotient of a -quasirandom finite group is again a -quasirandom finite group.)
• (ii) Conversely, if and are both -quasirandom, show that is -quasirandom also. (In particular, the direct or semidirect product of two -quasirandom finite groups is again a -quasirandom finite group.)

Informally, we will call quasirandom if it is -quasirandom for some “large” , though the precise meaning of “large” will depend on context. For applications to expansion in Cayley graphs, “large” will mean “ for some constant independent of the size of “, but other regimes of are certainly of interest.

The way we have set things up, the trivial group is infinitely quasirandom (i.e. it is -quasirandom for every ). This is however a degenerate case and will not be discussed further here. In the non-trivial case, a finite group can only be quasirandom if it is large and has no large subgroups:

Exercise 3 Let , and let be a finite -quasirandom group.

• (i) Show that if is non-trivial, then . (Hint: use the mean zero component of the regular representation .) In particular, non-trivial finite groups cannot be infinitely quasirandom.
• (ii) Show that any proper subgroup of has index . (Hint: use the mean zero component of the quasiregular representation.)

The following exercise shows that quasirandom groups have to be quite non-abelian, and in particular perfect:

Exercise 4 (Quasirandomness, abelianness, and perfection) Let be a finite group.

• (i) If is abelian and non-trivial, show that is not -quasirandom. (Hint: use Fourier analysis or the classification of finite abelian groups.)
• (ii) Show that is -quasirandom if and only if it is not perfect, i.e. the commutator group is a proper subgroup of . (Equivalently, is -quasirandom if and only if it has no non-trivial abelian quotients.)

Later on we shall see that there is a converse to the above two exercises; any non-trivial perfect finite group with no large subgroups will be quasirandom.

Exercise 5 Let be a finite -quasirandom group. Show that for any subgroup of , is -quasirandom, where is the index of in . (Hint: use induced representations.)

Now we give an example of a more quasirandom group.

Lemma 2 (Frobenius lemma) If is a field of some prime order , then is -quasirandom.

This should be compared with the cardinality of the special linear group, which is easily computed to be .

Proof: We may of course take to be odd. Suppose for contradiction that we have be a non-trivial representation on a unitary group of some dimension with . Set to be the group element

and suppose first that is non-trivial. Since , we have ; thus all the eigenvalues of are roots of unity. On the other hand, by conjugating by diagonal matrices in , we see that is conjugate to (and hence conjugate to ) whenever is a quadratic residue mod . As such, the eigenvalues of must be permuted by the operation for any quadratic residue mod . Since has at least one non-trivial eigenvalue, and there are distinct quadratic residues, we conclude that has at least distinct eigenvalues. But is a matrix with , a contradiction. Thus lies in the kernel of . By conjugation, we then see that this kernel contains all unipotent matrices. But these matrices generate (see exercise below), and so is trivial, a contradiction.

Exercise 6 Show that for any prime , the unipotent matrices

for any generate as a group.

Exercise 7 Let be a finite group, and let . If is generated by a collection of -quasirandom subgroups, show that is itself -quasirandom.

Exercise 8 Show that is -quasirandom for any and any prime . (This is not sharp; the optimal bound here is , which follows from the results of Landazuri and Seitz.)

As a corollary of the above results and Exercise 2, we see that the projective special linear group is also -quasirandom.

Remark 2 One can ask whether the bound in Lemma 2 is sharp, assuming of course that is odd. Noting that acts linearly on the plane , we see that it also acts projectively on the projective line , which has elements. Thus acts via the quasiregular representation on the -dimensional space , and also on the -dimensional subspace ; this latter representation (known as the Steinberg representation) is irreducible. This shows that the bound cannot be improved beyond . More generally, given any character , acts on the -dimensional space of functions that obey the twisted dilation invariance for all and ; these are known as the principal series representations. When is the trivial character, this is the quasiregular representation discussed earlier. For most other characters, this is an irreducible representation, but it turns out that when is the quadratic representation (thus taking values in while being non-trivial), the principal series representation splits into the direct sum of two -dimensional representations, which comes very close to matching the bound in Lemma 2. There is a parallel series of representations to the principal series (known as the discrete series) which is more complicated to describe (roughly speaking, one has to embed in a quadratic extension and then use a rotated version of the above construction, to change a split torus into a non-split torus), but can generate irreducible representations of dimension , showing that the bound in Lemma 2 is in fact exactly sharp. These constructions can be generalised to arbitrary finite groups of Lie type using Deligne-Luzstig theory, but this is beyond the scope of this course (and of my own knowledge in the subject).

Exercise 9 Let be an odd prime. Show that for any , the alternating group is -quasirandom. (Hint: show that all cycles of order in are conjugate to each other in (and not just in ); in particular, a cycle is conjugate to its power for all . Also, as , is simple, and so the cycles of order generate the entire group.)

Remark 3 By using more precise information on the representations of the alternating group (using the theory of Specht modules and Young tableaux), one can show the slightly sharper statement that is -quasirandom for (but is only -quasirandom for due to isocahedral symmetry, and -quasirandom for due to lack of perfectness). Using Exercise 3 with the index subgroup , we see that the bound cannot be improved. Thus, (for large ) is not as quasirandom as the special linear groups (for large and bounded), because in the latter case the quasirandomness is as strong as a power of the size of the group, whereas in the former case it is only logarithmic in size.

If one replaces the alternating group with the slightly larger symmetric group , then quasirandomness is destroyed (since , having the abelian quotient , is not perfect); indeed, is -quasirandom and no better.

Remark 4 Thanks to the monumental achievement of the classification of finite simple groups, we know that apart from a finite number (26, to be precise) of sporadic exceptions, all finite simple groups (up to isomorphism) are either a cyclic group , an alternating group , or is a finite simple group of Lie type such as . (We will define the concept of a finite simple group of Lie type more precisely in later notes, but suffice to say for now that such groups are constructed from reductive algebraic groups, for instance is constructed from in characteristic .) In the case of finite simple groups of Lie type with bounded rank , it is known from the work of Landazuri and Seitz that such groups are -quasirandom for some depending only on the rank. On the other hand, by the previous remark, the large alternating groups do not have this property, and one can show that the finite simple groups of Lie type with large rank also do not have this property. Thus, we see using the classification that if a finite simple group is -quasirandom for some and is sufficiently large depending on , then is a finite simple group of Lie type with rank . It would be of interest to see if there was an alternate way to establish this fact that did not rely on the classification, as it may lead to an alternate approach to proving the classification (or perhaps a weakened version thereof).

A key reason why quasirandomness is desirable for the purposes of demonstrating expansion is that quasirandom groups happen to be rapidly mixing at large scales, as we shall see below the fold. As such, quasirandomness is an important tool for demonstrating expansion in Cayley graphs, though because expansion is a phenomenon that must hold at all scales, one needs to supplement quasirandomness with some additional input that creates mixing at small or medium scales also before one can deduce expansion. As an example of this technique of combining quasirandomness with mixing at small and medium scales, we present a proof (due to Sarnak-Xue, and simplified by Gamburd) of a weak version of the famous “3/16 theorem” of Selberg on the least non-trivial eigenvalue of the Laplacian on a modular curve, which among other things can be used to construct a family of expander Cayley graphs in (compare this with the property (T)-based methods in the previous notes, which could construct expander Cayley graphs in for any fixed ).

— 1. Mixing in quasirandom groups —

Let be a finite group. Given two functions , we can define the convolution by the formula

This operation is bilinear and associative, but is not commutative unless is abelian. From the Cauchy-Schwarz inequality one has

and hence

This inequality is sharp in the sense that if we set and to both be constant-valued, then the left-hand side and right-hand side match. For abelian groups, one can also see this example is sharp when and are multiples of the same character.

It turns out, though, that if one restricts one of or (or both) to be of mean zero, and is quasirandom, then one can improve this inequality, which first appeared explicitly in the work of Babai, Nikolov, and Pyber:

Proposition 3 (Mixing inequality) Let be a finite -quasirandom group, and let . If at least one of has mean zero, then

Proof: By subtracting a constant from or , we may assume that or both have mean zero.

Observe that (being a superposition of right-translates of ) also has mean zero. Thus, we see that we may define an operator by setting . It thus suffices to show that the operator norm of is at most .

Fix . We can view as a matrix. We apply the singular value decomposition to this matrix to obtain singular values

of , together with associated singular vectors. The operator norm of is the largest singular value . The operator is then a self-adjoint operator (or matrix) with eigenvalues . In particular, we have

Now, a short computation shows that , where , and (by embedding in , and noting that annihilates constants) the trace can computed as

Thus, if is the eigenspace of corresponding to the eigenvalue (so that the dimension of is the multiplicity of ), we have

Now observe that if is the left-regular representation (restricted to ) then

for any and (this is a special case of the associativity of convolution). In particular, we see that is invariant under . Since has no non-trivial invariant vectors in , we conclude from quasirandomness that has dimension at least , and the claim follows.

Remark 5 One can also establish the above inequality using the nonabelian Fourier transform (which is based on the Peter-Weyl theorem combined with Schur’s lemma, and is developed in this blog post); we leave this as an exercise for the interested reader.

Exercise 10 Let be subsets of a finite -quasirandom group . Show that

and

Conclude in particular that

and that whenever . (The bounds here are not quite sharp, but are simpler than the optimal bounds, and suffice for most applications.)

Thus, for instance, if is a subset of a finite -quasirandom group of density more than , then will be most of (with fewer than elements omitted), and will be all of ; thus large subsets of a quasirandom group rapidly expand to fill out the whole group. In the converse direction, we have

Exercise 11 Let , and let be a finite group which is not -quasirandom. Show that there exists a subset of with for some absolute constant , such that . (Hint: by hypothesis, one has a non-trivial unitary representation of dimension at most . Show that contains an element at distance from the identity in operator norm, and take to be the preimage of a suitable ball around the identity in the operator norm.)

One can improve this result by using a quantitative form of Jordan’s theorem; see the next section.

Exercise 12 (Mixing inequality for actions) Let be a finite -quasirandom group acting (on the left) on a discrete set . Given functions and , one can define the convolution in much the same way as before:

Show that

whenever has mean zero, or whenever has mean zero on every orbit of .

One can use quasirandomness to show that Cayley graphs of very large degree in a quasirandom group are expanders:

Exercise 13 Let be a -regular Cayley graph in a finite -quasirandom group on vertices.

• (i) If , are subsets of , show that

  

  (compare with the expander mixing lemma, Exercise 9 from Notes 1).

• (ii) Show that is a two-sided -expander whenever

  

Unfortunately, the above result is only non-trivial in the regime , whereas for our applications we are interested instead in the regime when . We record a tool for this purpose.

Proposition 4 (Using quasirandomness to demonstrate expansion) Let be a -regular Cayley graph in a finite group . Assume the following:

• (i) (Quasirandomness) is -quasirandom for some .
• (ii) (Flattening of random walk) One has

  

  for some with and , where and is the -fold convolution of .

Then is a two-sided -expander for some depending only on , if is sufficiently large depending on these quantities. If we replace by in the flattening hypothesis, then is a one-sided -expander instead.

Proof: We allow implied constants to depend on . We will just prove the first claim, as the second claim is similar. By Exercise 5 from Notes 2, it will suffice to show that

(say) for some . But from Proposition 3 (with and ) and the hypotheses we have

for any . Iterating this (starting from, say, , and advancing in steps of times) we obtain the claim.

Exercise 14 Obtain an alternate proof of the above result that proceeds directly from the spectral decomposition of the adjacency operator into eigenvalues and eigenvectors and quasirandomness, rather than through Exercise 5 from Notes 2 and Proposition 3. (This alternate approach is closer in spirit to the arguments of Sarnak-Xue and Bourgain-Gamburd, though the two approaches are largely equivalent in the final analysis.)

Informally, the flattening hypothesis in Proposition 4 asserts that by time , the random walk has expanded to the point where it is covering a large portion of the group (roughly speaking, it is spread out over a set of size at least ). The point is that the scale of this set is large enough for the quasirandomness properties of the group to then mix the random walk rapidly towards the uniform distribution. However, this proposition provides no tools with which to prove this flattening property; this task will be a focus of subsequent notes.

The following exercise extends some of the above theory from quasirandom groups to “virtually quasirandom” groups, which have a bounded index subgroup that is quasirandom, but need not themselves be quasirandom.

Exercise 15 (Virtually quasirandom groups) Let be a finite group that contains a normal -quasirandom subgroup of index at most .

• (i) If , and at least one of has mean zero on every coset of , show that

  

• (ii) If for some , and is a connected -regular Cayley graph obeying (1) for some with and , show that is a two-sided -expander for some depending only on .

— 2. An algebraic description of quasirandomness (optional) —

As defined above, quasirandomness is a property of representations. However, one can reformulate this property (at a qualitative level, at least) in a more algebraic fashion, by means of Jordan’s theorem:

Theorem 5 (Jordan’s theorem) Let be a finite subgroup of for some . Then contains a normal abelian subgroup of index at most , where depends only on .

A proof of this theorem (giving a rather poor value of ) may be found in this previous blog post. The optimal value of is known for almost all , thanks to the classification of finite simple groups: for instance, it is a result of Collins that for (which is attained with the example of the symmetric group which acts on the space of -dimensional complex vectors whose coefficients sum to zero).

Jordan’s theorem can be used to give a qualitative description of quasirandomness, providing a converse to Exercises 3 and 4:

Exercise 16 Let be an integer. Let be a perfect finite group, with the property that all proper normal subgroups of have index greater than , where is the quantity in Jordan’s theorem. Show that is -quasirandom.

Conclude in particular that any finite simple nonabelian group of cardinality greater than is -quasirandom.

By using the classification of finite simple groups more carefully, Nikolov and Pyber were able to replace here by . Using related arguments, they also showed that if was not -quasirandom, then there was a subset of with cardinality such that , thus giving a reasonably tight converse to Exercise 10.

— 3. A weak form of Selberg’s 3/16 theorem (optional) —

Remark 6 This section presumes some familiarity with Riemannian geometry, as well as the functional analysis of Sobolev spaces and distributions. See for instance this blog post for a very brief introduction to Riemannian geometry, and these two previous posts for an introduction to distributions and Sobolev spaces.

We now give an application of quasirandomness to establish the following result, first observed explicitly by Lubotsky, Phillips and Sarnak as a corollary of a famous theorem of Selberg:

Theorem 6 (Selberg’s expander construction) If is a symmetric set of generators of that does not contain the identity or any elements of order , then the Cayley graphs form a one-sided expander family, where is the obvious projection homomorphism, and ranges over primes.

This is the analogue of Margulis’s expander construction (Corollary 13 from Notes 2), except that the modulus has been restricted here to be prime. This restriction can be removed with some additional effort, but we will not discuss this issue here. The condition that generates the entire group can be substantially relaxed; we will discuss this point in later notes.

In the property (T) approach to expansion, one passed from discrete groups such as to continuous groups such as , in order to take advantage of tools from analysis (such as limits). Similarly, to prove Theorem 6, we will pass from to . Actually, it will be convenient to work with the quotient space , better known as the hyperbolic plane. We will endow this plane with the structure of a Riemannian manifold, in order to access the Laplace-Beltrami operator on that plane, which is a continuous analogue (after some renormalisation) of the adjacency operator of a Cayley graph, which enjoys some nice exact identities which are difficult to discern in the discrete world.

We now therefore digress from the topic of expansion to recall the geometry of the hyperbolic plane. It will be convenient to switch between a number of different coordinatisations of this plane. Our primary model will be the half-plane model:

Definition 7 (Poincaré half-plane model) The Poincaré half-plane is the upper half-plane with the Riemannian metric . The (left) action of on this half-plane is given by the formula

One easily verifies that this gives an isometric action on .

Exercise 17 Verify that acts isometrically and transitively on , with stabiliser group isomorphic to ; thus is isomorphic (as an -homogeneous space) to .

Note that in some of the literature, a right action is used instead of a left action, leading to some reversals in the notational conventions used below, but this does not lead to any essential changes in the arguments or results.

Exercise 18 Show that the distance between two points is given by the formula

We can also use the model of the Poincaré disk with the Riemannian metric .

Exercise 19 Show that the Cayley transform is an isometric isomorphism from the half-plane to the disk .

Expressing an element of the Poincaré disk in exponential polar coordinates as , we can also model the Poincaré disk (in slightly singular coordinates) as the half-cylinder with metric . (Compare with the Euclidean plane in polar coordinates, which is similar but with the factor replaced by , or the sphere in Euler coordinates, which is also similar but with restricted to and replaced by . This similarity reflects the fact that these three Riemannian surfaces have constant curvature respectively.)

The action of can of course be described explicitly in the disk or half-plane models, but we will not need these explicit formulae here.

A Riemannian metric on a manifold always generates a measure on that manifold. For the Poincaré half-plane, the measure is . For the Poincaré disk, it is . For the half-cylinder, it is . In all cases, the action of will preserve the measure, because it preserves the metric, thus one can view as a Haar measure on the hyperbolic plane.

A Riemannian metric also generates a Laplace-Beltrami operator . In the Poincaré half-plane model, it is

in the Poincaré disk model, it is

and in the half-cylinder model, it is

Again, in all cases, the Laplacian will commute with the action of , because this action preserves the metric.

The discrete group acts on the hyperbolic plane, giving rise to a quotient , known as the principal modular curve of level . This quotient can also be viewed by taking the (closure of a) fundamental domain

and then identifying with on the left and right sides of this domain, and also identifying with on the lower boundary of this domain. The quotient is not compact, but it does have finite measure with respect to ; indeed, outside of a compact set, behaves like the cusp

for any constant , again identifying the boundary with the boundary, and this strip has measure

Thus descends to a finite Haar measure on .

The quotient is not quite a smooth Riemannian manifold, due to the presence of partially fixed points of the action at and (of order and order respectively), and is thus technically an orbifold rather than a manifold. However, this distinction turns out to not significantly affect the analysis and will be glossed over here.

The Laplace-Beltrami operator is defined on smooth compactly supported functions on , and then descends to an operator on smooth compactly supported functions on . (Here, we use the smooth structure on inherited from , thus a function is smooth at a point in if it lifts to a function smooth at the preimage of that point.) On , we have the integration by parts formula

where is the gradient with respect to the Riemannian metric, and the inner product; in the half-plane coordinates, we have

in the disk model, it is

and in the half-cylinder model, it is

In particular, we have the positive definiteness property

for all . This descends to :

Thus is a symmetric positive-definite densely-defined operator on . One can in fact show (by solving some PDE, such as the wave equation or the resolvent equation, and exploiting at some point the fact that the Riemannian manifold is complete) that is essentially self-adjoint and is thus subject to the spectral theorem, but we will avoid using the full force of spectral theory here.

Since has finite measure and , we see that is an eigenfunction of (or ) with eigenvalue zero. We eliminate this eigenfunction by working in the space (or ) of functions in (or ) of mean zero. Let us now define the spectral gap to be the quantity

Then . Using the spectral theorem, one can interpret the spectral gap as the infimum of the spectrum of the (negative) Laplacian on . Note also that one can take to be either real or complex valued, as this will not affect the value of the spectral gap. Also by a truncation and mollification argument we may allow to range in instead of here if desired.

We have the following bounds:

Proposition 8 (Spectral gap of ) We have .

Proof: We first establish the upper bound . It will suffice to find non-zero functions whose Rayleigh quotient

is arbitrarily close to .

We will restrict attention to smooth compactly supported functions supported on the cusp (2) for a fixed (e.g. one can take ). In coordinates, the Raleigh quotient becomes

where we use subscripts to denote partial differentiation, while the mean zero condition becomes

To build such functions, we select a large parameter , and choose a function that depends only on the variable, is supported on the region , and equals in the region and is smoothly truncated in the intermediate region (assigning enough negative mass in the region to obtain the mean zero condition). A brief calculation shows that

and

(where the implied constant can depend on but not on ), and so the claim follows by sending .

Now we show the lower bound . Suppose this claim failed; then we may find a sequence of functions with such that

where denotes a quantity that goes to zero as . We can take the to be real valued.

To deal with the non-compact portion of (i.e. the strip (2)) we now use Hardy’s inequality. Observe that if is smooth, real-valued, and compactly supported on a cusp (2) for some (we can take as before), then by integration by parts

and hence by Cauchy-Schwarz

Applying this to a truncated version of for some and some smooth cutoff supported on that equals one on , we conclude that

For any , one can use the pigeonhole principle to find an (depending on ) such that

and thus we see that

for some depending only on . Thus, the probability measures form a tight sequence of measures in . As the are also locally uniformly bounded in the Sobolev space , we conclude from the Rellich compactness theorem (or the Poincaré inequality) that after passing to a subsequence, the converge strongly in to a limit , which then has norm one, mean zero, and in a distributional sense. But then by the Poincaré inequality, is constant, which is absurd.

Remark 7 One can in fact establish after some calculation using the theory of modular forms that is exactly , but we will not do so here. By modifying the above arguments, one can in fact show that on has absolutely continuous spectrum on .

Now we move back towards the task of establishing expansion for the Cayley graphs . Let denote the kernel of the projection map ; this is the group of matrices in that are equal to mod , and is known as a principal congruence subgroup of . It is a finite index normal subgroup of . In analogy with what we did for , we can then define the principal modular curve , and then define the Laplacian on this curve and the spectral gap . At a qualitative level, the geometry of is similar to that of , except that instead of having just one cusp (2), there are now multiple cusps (which do not necessarily go to infinity as in (2), but may instead go to some other point on the boundary of the hyperbolic plane). (One may think of as being formed by cutting up a finite number of of ‘s and then randomly sowing them together to create a tangled orbifold that is a continuous analogue of an expander graph; see this Notices article of Sarnak for more discussion of this perspective.) By a routine modification of Proposition 8, one can show that

(Note also that as is a finite isometric cover of , we have the trivial bound .) However, these arguments do not keep uniformly bounded away from zero. Much more is conjectured to be true:

Conjecture 9 (Selberg’s conjecture) One has for all (not necessarily prime).

This conjecture remains open (though it has been verified numerically for small values of , in particular for all by Booker and Stromgbergsson). On the other hand, we have the following celebrated result of Selberg:

Theorem 10 (Selberg’s 3/16 theorem) One has for all (not necessarily prime).

Selberg’s argument uses a serious amount of number-theoretic machinery (in particular, bounds for Kloosterman sums) and will not be reproduced here. The bound has since been improved; the best bound currentlt known is , due to Kim and Sarnak and involving even more number-theoretic machinery (related to the Langlands conjectures); this argument will also not be discussed further here.

In the papers of by Sarnak and Xue and by Gamburd, an argument based primarily on quasirandomness that used only very elementary number theory was introduced, to obtain the following result:

Theorem 11 (Weak Selberg theorem) One has for all primes , where goes to zero as .

In particular, one has a uniform lower bound for some absolute constant (and, since one can compute that , one can in fact take ). Despite giving a weaker result than Theorem 10, the argument is more flexible and can be applied to other arithmetic surfaces than , for which the method of Selberg does not seem to apply; see the papers of Sarnak-Xue and Gamburd for further discussion.

We will not quite prove Theorem 11 here, but instead establish the following even weaker version which uses the same ideas, but in a slightly less computation-intensive fashion (at the cost of some efficiency in the argument):

Theorem 12 (Even weaker Selberg theorem) One has for all primes .

Of course, this result is still strong enough to supply a uniform lower bound on .

Before we prove Theorem 12 (a spectral gap in the continuous world), let us show how it can be transferred to deduce Theorem 6 (a spectral gap in the discrete world). Suppose for contradiction that Theorem 6 failed. Then we can find a finite symmetric generating set for (with no elements of order one or two) and a sequence of primes going to infinity such that the one-sided expansion constant of goes to zero. Write . Applying the weak discrete Cheeger inequality (Exercise 3 from Notes 2), we conclude that we can find non-empty subsets of size which are almost -invariant in the sense that that . Since generates , we conclude in particular that

for any independent of .

The idea now is to pass from this nearly-invariant discrete set to a nearly-invariant continuous analogue to which the uniform bound on the spectral gap can be applied to obtain a contradiction. (This argument is similar in spirit to Proposition 9 from Notes 2.)

We turn to the details. Let be a large parameter (independent of ) to be chosen later, and let be a point on (avoiding fixed points of the action); for sake of concretness we can take to be the projection of to . Note that acts on . We consider the function defined by the formula

This function equals when is within (in the hyperbolic metric) of a point in the orbit , and equals when is further than of this orbit; in particular, it is compact supported. The function is also Lipschitz with constant , so (using a weak derivative).

The curve is a -fold cover of and thus has volume . Observe that equals on the -neighbourhood of any point in . As these points are separated from each other by a bounded distance (independent of and ), we conclude that

where the bound is uniform in . Conversely, if is not of the form for some and some within distance from the identity, we have equal to on the -neighbourhood of . There are only possible choices for ; since is independent of , we conclude from (3) that all but in are of the form described above. Since , we conclude that

if is sufficiently large depending on . As a consequence, if we let

be the mean-free component of , we have the lower bound

for sufficiently large depending on .

On the other hand, is non-zero only at points which are at distance between and of . Call the set of such points . To estimate the volume , we partition into sets of the form , where is a fundamental domain of (projected onto ) and ranges over . Because the ball of radius centred at is precompact and thus meets only of the translated domains , we see that the only for which meets are of the form , where lies in and lies in a subset of of size that is independent of . From (3) we conclude that all but at most of these thus lie in , and so

Since , we conclude that

for sufficiently large depending on . But this and (4) contradict the uniform lower bound on the spectral gap (after regularising in a standard fashion to make it smooth rather than merely Lipschitz), giving the desired contradiction.

We now turn to the proof of Theorem 12. The first step is to show that the only source of spectrum below is provided by eigenfunctions.

Proposition 13 (Discrete spectrum below ) Suppose that . Then there exists a non-zero such that (in the distributional sense).

Note that while is only initially in , it is a routine application of elliptic regularity (which we omit here) to show that is necessarily smooth.

Proof: For notational simplicity, we will just prove the claim in the case, though the general case is similar. Write , so that . The argument will be similar in spirit to the proof of the lower bound . Indeed, by definition of , we can find a sequence of functions with such that

As before, we can take the to be real valued.

Using Hardy’s inequality as in the proof of Proposition 8, we see that

for any . For any , one can use the pigeonhole principle to find an (depending on ) such that

and thus we see that

for some depending only on . If is small enough, , and thus

for all sufficiently large . By (5), is also uniformly bounded in norm. Thus by the Rellich compactness theorem, we may pass to a subsequence and assume that converges weakly in and strongly in to a limit , which is then non-zero. Also, from (6) we see that for each and there is an such that

and thus

Taking limits in weak (and strong ), we conclude that for some larger than that

Sending using monotone convergence, we conclude that

for any ; by definition of , we must then have

Perturbing in some test function direction of mean zero, and using the definition of , we conclude that

for all such . The mean zero condition on can be removed since both sides of this equation vanish when is constant. By duality we thus see that in the sense of distributions, as required.

Exercise 20 Establish the above proposition for general .

Exercise 21 Show that for any , the spectrum of in on is finite (and in particular consists only of eigenvalues), with each eigenvalue having finite multiplicity. (For this exercise you may use without proof the fact that is essentially self-adjoint.)

We will also need a variant of the above proposition:

Lemma 14 (Eigenfunctions do not concentrate in cusps) Let . Then there is a compact subset of , such that for any and any eigenfunction on with some , one has

where the implied constant is independent of , , and , and is the covering map.

The lemma is basically proved by applying Hardy’s inequality to each cusp of ; see the paper of Gamburd for details.

Now we can start using quasirandomness. Let be the space of all eigenfunctions of of eigenvalue :

By the above proposition, this is a non-trivial Hilbert space. From Exercise 21, is finite-dimensional (though we do not really need to know this fact yet in the argument that follows, as it will be a consequence of the computations). Since acts isometrically on , it also acts on . If is a -invariant vector in , then it descends to a function on , which is impossible if . Applying the Frobenius lemma (Lemma 2), we conclude

Lemma 15 (Quasirandomness) If , then has dimension at least .

To complement this quasirandomness to get expansion, we need a flattening property, as in Proposition 4. In the discrete world, we applied a flattening property to the distribution of a long discrete random walk. The direct analogue of such a distribution would be a heat kernel of the Laplacian , and this is what we shall use here. (It turns out that the heat kernel is not quite the most efficient object to analyse here; see Remark 10 below.)

We first recall the formula for the heat kernel on the hyperbolic plane (which can be found in many places, such as this text of Chavel, or this text of Terras):

Exercise 22 Show that the heat operator on test functions in is given by the formula

where is the kernel

(Hint: There are two main computations. One is to show that obeys the heat equation, which in half-cylindrical coordinates means that one has to verify that

The other is to show that resembles the Euclidean heat kernel for small . There are several other ways to derive this formula in terms of formulae for other operators (e.g. the wave propagator); see for instance Terras’s book for some discussion.)

For our purposes, we only need a crude upper bound on the heat kernel:

Exercise 23 With the notation of the preceding section, show that

when and .

In our applications, the polynomial factors will be negligible; only the exponential factors will be of importance. Note that if one integrates the above estimate against the measure , one sees that

The right-hand side evaluates to . On the other hand, as the heat kernel is a probability measure, one has . Thus, up to polynomial factors, the above estimate is quite tight.

Remark 8 From Exercise 23, we see that the probability measure concentrates around the region ; thus on the hyperbolic plane, Brownian motion moves “ballistically” away from its starting point at a unit speed, in contrast to the situation in Euclidean geometry, where after time a Brownian motion is only expected to move by a distance . One can see this phenomenon also from the heat equation (7), which when expressed in terms of the probability density becomes a Fokker-Planck equation

with unit diffusion and drift speed . Since rapidly approaches when becomes large, we thus expect to concentrate in the region , as is indeed the case.

We let be a parameter to optimise in later. The heat operator on descends to a heat operator on the quotient , defined by the formula

for ; note that the sum is -invariant, and so makes sense for and not just for . When applied to an eigenfunction , one has ; in particular, preserves , and thus also preserves the orthogonal complement of . As is positive semi-definite, it therefore splits as the sum of and another positive semi-definite operator, where is the orthogonal projection to . Note that is an integral operator with kernel

where is an orthonormal basis of . (Here we use the fact that is finite dimensional, but if one does not want to use this fact yet, one can work instead with a finite-dimensional subspace of in the argument that follows.) Since positive semi-definite integral operators (with continuous kernel) are non-negative on the diagonal, we conclude the pointwise inequality

for all .

This will be our starting point to get a lower bound on . But first we must deal with the other quantities , in this expression. A simple way to proceed here is to integrate out in to exploit the normalisation of the :

However, this turns out to be a little unfavourable because the integrand on the right-hand side does not behave well enough at cusps. However, if one uses Lemma 14 first (assuming that ), and integrates over the resulting region , we can avoid the cusps:

Because the sum here is -invariant, we can descend from to :

We now insert the bound in Exercise 23, as well as the bound :

Because is compact, we can use the triangle inequality to bound

(say). This is convenient, because is computable:

Exercise 24 For any , show that , where is the Frobenius norm of the matrix .

Inserting these bounds, we obtain

Decomposing according to the integer part of , we thus have

where is the counting function

So one is left with the purely number-theoretic task of estimating . This is basically the number of points of in the ball of radius in . From the half-cylinder model, we see that the measure of this ball is . On the other hand, has index in , which has bounded covolume in . We thus heuristically expect to be . If this were the truth, then the right-hand side of (9) would be (cf. the evaluation of (8)), which when combined with quasirandomness (Lemma 15) would give a lower bound of , that would be particularly strong when was small.

The key is then the following “flattening lemma”, that shows that is indeed roughly of the order of when is large, and is the main number-theoretic input to the argument:

Lemma 16 (Flattening lemma) For any , one has

Proof: Using the definition of , we are basically counting the number of integer solutions to the equation

subject to the congruences

and the bounds

Since are both divisible by , we see also that . Similarly, as and are divisible by , we have . Subtracting, we conclude that

Now we proceed as follows. The number of integers with is . For each such , the number of with and is . For each fixed and , the expression is of size ; by the divisor bound, there are thus ways to factor into . Thus, we obtain a final bound of

giving the claim.

Remark 9 One can obtain improved bounds to for some ranges of (particularly when ranges between and ) by using more advanced tools, such as bounds on Kloosterman sums. Unfortunately, such improvements do not actually improve the final constants in this argument. (Kloosterman sums do however play a key role in the proof of Theorem 10, which proceeds by a different, and more highly arithmetic, argument.)

A routine calculation then finishes off the proof of Theorem 12:

Exercise 25 Using the above lemma, show that the right-hand side of (9) is

for any . Optimising this in and using Lemma 15, establish a contradiction whenever and is sufficiently large depending on , thus giving Theorem 12.

Remark 10 An inspection of the above argument shows that the term in (10) is the main obstacle to improving the constant. This term ultimately can be “blamed” for the relatively large value of the heat kernel at the origin. To improve this, one can observe that the main features of the heat kernel that were needed for the argument were that it was positive definite, and had an explicit effect on eigenfunctions . It turns out that there are several other kernels with these properties, and by selecting a kernel with less concentration at the identity, one can obtain a better result. In particular, an efficient choice of kernel turns out to be the convolution of a ball of radius with itself. By performing some additional calculations in hyperbolic geometry (in particular, using the Selberg/Harish-Chandra theory of spherical functions) one can use this kernel to improve the bound given here to ; see the paper of Gamburd for details. Unfortunately, the fraction here appears to be the limit of this particular method.

For reasons of space, this paper is very far from an exhaustive survey even of the narrow topic of universality for Wigner matrices, but should hopefully be an accessible entry point into the subject nevertheless.