Theorem 1 (Eigenvector-eigenvalue identity)Let be an Hermitian matrix, with eigenvalues . Let be a unit eigenvector corresponding to the eigenvalue , and let be the component of . Thenwhere is the Hermitian matrix formed by deleting the row and column from .

When we posted the first version of this paper, we were unaware of previous appearances of this identity in the literature; a related identity had been used by Erdos-Schlein-Yau and by myself and Van Vu for applications to random matrix theory, but to our knowledge this specific identity appeared to be new. Even two months after our preprint first appeared on the arXiv in August, we had only learned of one other place in the literature where the identity showed up (by Forrester and Zhang, who also cite an earlier paper of Baryshnikov).

The situation changed rather dramatically with the publication of a popular science article in Quanta on this identity in November, which gave this result significantly more exposure. Within a few weeks we became informed (through private communication, online discussion, and exploration of the citation tree around the references we were alerted to) of over three dozen places where the identity, or some other closely related identity, had previously appeared in the literature, in such areas as numerical linear algebra, various aspects of graph theory (graph reconstruction, chemical graph theory, and walks on graphs), inverse eigenvalue problems, random matrix theory, and neutrino physics. As a consequence, we have decided to completely rewrite our article in order to collate this crowdsourced information, and survey the history of this identity, all the known proofs (we collect seven distinct ways to prove the identity (or generalisations thereof)), and all the applications of it that we are currently aware of. The citation graph of the literature that this *ad hoc* crowdsourcing effort produced is only very weakly connected, which we found surprising:

The earliest explicit appearance of the eigenvector-eigenvalue identity we are now aware of is in a 1966 paper of Thompson, although this paper is only cited (directly or indirectly) by a fraction of the known literature, and also there is a precursor identity of Löwner from 1934 that can be shown to imply the identity as a limiting case. At the end of the paper we speculate on some possible reasons why this identity only achieved a modest amount of recognition and dissemination prior to the November 2019 Quanta article.

]]>Let be a discrete group. A *(concrete) measure-preserving action* of on is a group homomorphism from to , thus is the identity map and for all . A large portion of ergodic theory is concerned with the study of such measure-preserving actions, especially in the classical case when is the integers (with the additive group law).

Let be a compact Hausdorff abelian group, which we can endow with the Borel -algebra . A *(concrete measurable) –cocycle* is a collection of concrete measurable maps obeying the *cocycle equation*

for -almost every . (Here we are glossing over a measure-theoretic subtlety that we will return to later in this post – see if you can spot it before then!) Cocycles arise naturally in the theory of group extensions of dynamical systems; in particular (and ignoring the aforementioned subtlety), each cocycle induces a measure-preserving action on (which we endow with the product of with Haar probability measure on ), defined by

This connection with group extensions was the original motivation for our study of measurable cohomology, but is not the focus of the current paper.

A special case of a -valued cocycle is a *(concrete measurable) -valued coboundary*, in which for each takes the special form

for -almost every , where is some measurable function; note that (ignoring the aforementioned subtlety), every function of this form is automatically a concrete measurable -valued cocycle. One of the first basic questions in measurable cohomology is to try to characterize which -valued cocycles are in fact -valued coboundaries. This is a difficult question in general. However, there is a general result of Moore and Schmidt that at least allows one to reduce to the model case when is the unit circle , by taking advantage of the Pontryagin dual group of characters , that is to say the collection of continuous homomorphisms to the unit circle. More precisely, we have

Theorem 1 (Countable Moore-Schmidt theorem)Let be a discrete group acting in a concrete measure-preserving fashion on a probability space . Let be a compact Hausdorff abelian group. Assume the following additional hypotheses:

- (i) is at most countable.
- (ii) is a standard Borel space.
- (iii) is metrisable.
Then a -valued concrete measurable cocycle is a concrete coboundary if and only if for each character , the -valued cocycles are concrete coboundaries.

The hypotheses (i), (ii), (iii) are saying in some sense that the data are not too “large”; in all three cases they are saying in some sense that the data are only “countably complicated”. For instance, (iii) is equivalent to being second countable, and (ii) is equivalent to being modeled by a complete separable metric space. It is because of this restriction that we refer to this result as a “countable” Moore-Schmidt theorem. This theorem is a useful tool in several other applications, such as the Host-Kra structure theorem for ergodic systems; I hope to return to these subsequent applications in a future post.

Let us very briefly sketch the main ideas of the proof of Theorem 1. Ignore for now issues of measurability, and pretend that something that holds almost everywhere in fact holds everywhere. The hard direction is to show that if each is a coboundary, then so is . By hypothesis, we then have an equation of the form

for all and some functions , and our task is then to produce a function for which

for all .

Comparing the two equations, the task would be easy if we could find an for which

for all . However there is an obstruction to this: the left-hand side of (3) is additive in , so the right-hand side would have to be also in order to obtain such a representation. In other words, for this strategy to work, one would have to first establish the identity

for all . On the other hand, the good news is that if we somehow manage to obtain the equation, then we can obtain a function obeying (3), thanks to Pontryagin duality, which gives a one-to-one correspondence between and the homomorphisms of the (discrete) group to .

Now, it turns out that one cannot derive the equation (4) directly from the given information (2). However, the left-hand side of (2) is additive in , so the right-hand side must be also. Manipulating this fact, we eventually arrive at

In other words, we don’t get to show that the left-hand side of (4) vanishes, but we do at least get to show that it is -invariant. Now let us assume for sake of argument that the action of is ergodic, which (ignoring issues about sets of measure zero) basically asserts that the only -invariant functions are constant. So now we get a weaker version of (4), namely

for some constants .

Now we need to eliminate the constants. This can be done by the following group-theoretic projection. Let denote the space of concrete measurable maps from to , up to almost everywhere equivalence; this is an abelian group where the various terms in (5) naturally live. Inside this group we have the subgroup of constant functions (up to almost everywhere equivalence); this is where the right-hand side of (5) lives. Because is a divisible group, there is an application of Zorn’s lemma (a good exercise for those who are not acquainted with these things) to show that there exists a retraction , that is to say a group homomorphism that is the identity on the subgroup . We can use this retraction, or more precisely the complement , to eliminate the constant in (5). Indeed, if we set

then from (5) we see that

while from (2) one has

and now the previous strategy works with replaced by . This concludes the sketch of proof of Theorem 1.

In making the above argument rigorous, the hypotheses (i)-(iii) are used in several places. For instance, to reduce to the ergodic case one relies on the ergodic decomposition, which requires the hypothesis (ii). Also, most of the above equations only hold outside of a set of measure zero, and the hypothesis (i) and the hypothesis (iii) (which is equivalent to being at most countable) to avoid the problem that an uncountable union of sets of measure zero could have positive measure (or fail to be measurable at all).

My co-author Asgar Jamneshan and I are working on a long-term project to extend many results in ergodic theory (such as the aforementioned Host-Kra structure theorem) to “uncountable” settings in which hypotheses analogous to (i)-(iii) are omitted; thus we wish to consider actions on uncountable groups, on spaces that are not standard Borel, and cocycles taking values in groups that are not metrisable. Such uncountable contexts naturally arise when trying to apply ergodic theory techniques to combinatorial problems (such as the inverse conjecture for the Gowers norms), as one often relies on the ultraproduct construction (or something similar) to generate an ergodic theory translation of these problems, and these constructions usually give “uncountable” objects rather than “countable” ones. (For instance, the ultraproduct of finite groups is a hyperfinite group, which is usually uncountable.). This paper marks the first step in this project by extending the Moore-Schmidt theorem to the uncountable setting.

If one simply drops the hypotheses (i)-(iii) and tries to prove the Moore-Schmidt theorem, several serious difficulties arise. We have already mentioned the loss of the ergodic decomposition and the possibility that one has to control an uncountable union of null sets. But there is in fact a more basic problem when one deletes (iii): the addition operation , while still continuous, can fail to be measurable as a map from to ! Thus for instance the sum of two measurable functions need not remain measurable, which makes even the very definition of a measurable cocycle or measurable coboundary problematic (or at least unnatural). This phenomenon is known as the *Nedoma pathology*. A standard example arises when is the uncountable torus , endowed with the product topology. Crucially, the Borel -algebra generated by this uncountable product is *not* the product of the factor Borel -algebras (the discrepancy ultimately arises from the fact that topologies permit uncountable unions, but -algebras do not); relating to this, the product -algebra is *not* the same as the Borel -algebra , but is instead a strict sub-algebra. If the group operations on were measurable, then the diagonal set

would be measurable in . But it is an easy exercise in manipulation of -algebras to show that if are any two measurable spaces and is measurable in , then the fibres of are contained in some countably generated subalgebra of . Thus if were -measurable, then all the points of would lie in a single countably generated -algebra. But the cardinality of such an algebra is at most while the cardinality of is , and Cantor’s theorem then gives a contradiction.

To resolve this problem, we give a coarser -algebra than the Borel -algebra, which we call the *reduced -algebra* , thus coarsening the measurable space structure on to a new measurable space . In the case of compact Hausdorff abelian groups, can be defined as the -algebra generated by the characters ; for more general compact abelian groups, one can define as the -algebra generated by all continuous maps into metric spaces. This -algebra is equal to when is metrisable but can be smaller for other . With this measurable structure, becomes a measurable group; it seems that once one leaves the metrisable world that is a superior (or at least equally good) space to work with than for analysis, as it avoids the Nedoma pathology. (For instance, from Plancherel’s theorem, we see that if is the Haar probability measure on , then (thus, every -measurable set is equivalent modulo -null sets to a -measurable set), so there is no damage to Plancherel caused by passing to the reduced -algebra.

Passing to the reduced -algebra fixes the most severe problems with an uncountable Moore-Schmidt theorem, but one is still faced with an issue of having to potentially take an uncountable union of null sets. To avoid this sort of problem, we pass to the framework of *abstract measure theory*, in which we remove explicit mention of “points” and can easily delete all null sets at a very early stage of the formalism. In this setup, the category of concrete measurable spaces is replaced with the larger category of *abstract measurable spaces*, which we formally define as the opposite category of the category of -algebras (with Boolean algebra homomorphisms). Thus, we define an *abstract measurable space* to be an object of the form , where is an (abstract) -algebra and is a formal placeholder symbol that signifies use of the opposite category, and an *abstract measurable map* is an object of the form , where is a Boolean algebra homomorphism and is again used as a formal placeholder; we call the *pullback map* associated to . [UPDATE: It turns out that this definition of a measurable map led to technical issues. In a forthcoming revision of the paper we also impose the requirement that the abstract measurable map be -complete (i.e., it respects countable joins).] The composition of two abstract measurable maps , is defined by the formula , or equivalently .

Every concrete measurable space can be identified with an abstract counterpart , and similarly every concrete measurable map can be identified with an abstract counterpart , where is the pullback map . Thus the category of concrete measurable spaces can be viewed as a subcategory of the category of abstract measurable spaces. The advantage of working in the abstract setting is that it gives us access to more spaces that could not be directly defined in the concrete setting. Most importantly for us, we have a new abstract space, the *opposite measure algebra* of , defined as where is the ideal of null sets in . Informally, is the space with all the null sets removed; there is a canonical abstract embedding map , which allows one to convert any concrete measurable map into an abstract one . One can then define the notion of an abstract action, abstract cocycle, and abstract coboundary by replacing every occurrence of the category of concrete measurable spaces with their abstract counterparts, and replacing with the opposite measure algebra ; see the paper for details. Our main theorem is then

Theorem 2 (Uncountable Moore-Schmidt theorem)Let be a discrete group acting abstractly on a -finite measure space . Let be a compact Hausdorff abelian group. Then a -valued abstract measurable cocycle is an abstract coboundary if and only if for each character , the -valued cocycles are abstract coboundaries.

With the abstract formalism, the proof of the uncountable Moore-Schmidt theorem is almost identical to the countable one (in fact we were able to make some simplifications, such as avoiding the use of the ergodic decomposition). A key tool is what we call a “conditional Pontryagin duality” theorem, which asserts that if one has an abstract measurable map for each obeying the identity for all , then there is an abstract measurable map such that for all . This is derived from the usual Pontryagin duality and some other tools, most notably the completeness of the -algebra of , and the Sikorski extension theorem.

We feel that it is natural to stay within the abstract measure theory formalism whenever dealing with uncountable situations. However, it is still an interesting question as to when one can guarantee that the abstract objects constructed in this formalism are representable by concrete analogues. The basic questions in this regard are:

- (i) Suppose one has an abstract measurable map into a concrete measurable space. Does there exist a representation of by a concrete measurable map ? Is it unique up to almost everywhere equivalence?
- (ii) Suppose one has a concrete cocycle that is an abstract coboundary. When can it be represented by a concrete coboundary?

For (i) the answer is somewhat interesting (as I learned after posing this MathOverflow question):

- If does not separate points, or is not compact metrisable or Polish, there can be counterexamples to uniqueness. If is not compact or Polish, there can be counterexamples to existence.
- If is a compact metric space or a Polish space, then one always has existence and uniqueness.
- If is a compact Hausdorff abelian group, one always has existence.
- If is a complete measure space, then one always has existence (from a theorem of Maharam).
- If is the unit interval with the Borel -algebra and Lebesgue measure, then one has existence for all compact Hausdorff assuming the continuum hypothesis (from a theorem of von Neumann) but existence can fail under other extensions of ZFC (from a theorem of Shelah, using the method of forcing).
- For more general , existence for all compact Hausdorff is equivalent to the existence of a lifting from the -algebra to (or, in the language of abstract measurable spaces, the existence of an abstract retraction from to ).
- It is a long-standing open question (posed for instance by Fremlin) whether it is relatively consistent with ZFC that existence holds whenever is compact Hausdorff.

Our understanding of (ii) is much less complete:

- If is metrisable, the answer is “always” (which among other things establishes the countable Moore-Schmidt theorem as a corollary of the uncountable one).
- If is at most countable and is a complete measure space, then the answer is again “always”.

In view of the answers to (i), I would not be surprised if the full answer to (ii) was also sensitive to axioms of set theory. However, such set theoretic issues seem to be almost completely avoided if one sticks with the abstract formalism throughout; they only arise when trying to pass back and forth between the abstract and concrete categories.

]]>The basic objects of study in analytic number theory are deterministic; there is nothing inherently random about the set of prime numbers, for instance. Despite this, one can still interpret many of the averages encountered in analytic number theory in probabilistic terms, by introducing random variables into the subject. Consider for instance the form

of the prime number theorem (where we take the limit ). One can interpret this estimate probabilistically as

where is a random variable drawn uniformly from the natural numbers up to , and denotes the expectation. (In this set of notes we will use boldface symbols to denote random variables, and non-boldface symbols for deterministic objects.) By itself, such an interpretation is little more than a change of notation. However, the power of this interpretation becomes more apparent when one then imports concepts from probability theory (together with all their attendant intuitions and tools), such as independence, conditioning, stationarity, total variation distance, and entropy. For instance, suppose we want to use the prime number theorem (1) to make a prediction for the sum

After dividing by , this is essentially

With probabilistic intuition, one may expect the random variables to be approximately independent (there is no obvious relationship between the number of prime factors of , and of ), and so the above average would be expected to be approximately equal to

which by (2) is equal to . Thus we are led to the prediction

The asymptotic (3) is widely believed (it is a special case of the *Chowla conjecture*, which we will discuss in later notes; while there has been recent progress towards establishing it rigorously, it remains open for now.

How would one try to make these probabilistic intuitions more rigorous? The first thing one needs to do is find a more quantitative measurement of what it means for two random variables to be “approximately” independent. There are several candidates for such measurements, but we will focus in these notes on two particularly convenient measures of approximate independence: the “” measure of independence known as covariance, and the “” measure of independence known as mutual information (actually we will usually need the more general notion of conditional mutual information that measures conditional independence). The use of type methods in analytic number theory is well established, though it is usually not described in probabilistic terms, being referred to instead by such names as the “second moment method”, the “large sieve” or the “method of bilinear sums”. The use of methods (or “entropy methods”) is much more recent, and has been able to control certain types of averages in analytic number theory that were out of reach of previous methods such as methods. For instance, in later notes we will use entropy methods to establish the logarithmically averaged version

of (3), which is implied by (3) but strictly weaker (much as the prime number theorem (1) implies the bound , but the latter bound is much easier to establish than the former).

As with many other situations in analytic number theory, we can exploit the fact that certain assertions (such as approximate independence) can become significantly easier to prove if one only seeks to establish them *on average*, rather than uniformly. For instance, given two random variables and of number-theoretic origin (such as the random variables and mentioned previously), it can often be extremely difficult to determine the extent to which behave “independently” (or “conditionally independently”). However, thanks to second moment tools or entropy based tools, it is often possible to assert results of the following flavour: if are a large collection of “independent” random variables, and is a further random variable that is “not too large” in some sense, then must necessarily be nearly independent (or conditionally independent) to many of the , even if one cannot pinpoint precisely which of the the variable is independent with. In the case of the second moment method, this allows us to compute correlations such as for “most” . The entropy method gives bounds that are significantly weaker quantitatively than the second moment method (and in particular, in its current incarnation at least it is only able to say non-trivial assertions involving interactions with residue classes at small primes), but can control significantly more general quantities for “most” thanks to tools such as the Pinsker inequality.

** — 1. Second moment methods — **

In this section we discuss probabilistic techniques of an “” nature. We fix a probability space to model all of random variables; thus for instance we shall model a complex random variable in these notes by a measurable function . (Strictly speaking, there is a subtle distinction one can maintain between a random variable and its various measure-theoretic models, which becomes relevant if one later decides to modify the probability space , but this distinction will not be so important in these notes and so we shall ignore it. See this previous set of notes for more discussion.)

We will focus here on the space of complex random variables (that is to say, measurable maps ) whose *second moment*

of is finite. In many number-theoretic applications the finiteness of the second moment will be automatic because will only take finitely many values. As is well known, the space has the structure of a complex Hilbert space, with inner product

and norm

for . By slight abuse of notation, the complex numbers can be viewed as a subset of , by viewing any given complex number as a constant (deterministic) random variable. Then is a one-dimensional subspace of , spanned by the unit vector . Given a random variable to , the projection of to is then the *mean*

and we obtain an orthogonal splitting of any into its mean and its mean zero part . By Pythagoras’ theorem, we then have

The first quantity on the right-hand side is the square of the distance from to , and this non-negative quantity is known as the variance

The square root of the variance is known as the standard deviation. The variance controls the distribution of the random variable through Chebyshev’s inequality

for any , which is immediate from observing the inequality and then taking expectations of both sides. Roughly speaking, this inequality asserts that typically deviates from its mean by no more than a bounded multiple of the standard deviation .

A slight generalisation of Chebyshev’s inequality that can be convenient to use is

for any and any complex number (which typically will be a simplified approximation to the mean ), which is proven similarly to (6) but noting (from (5)) that .

Informally, (6) is an assertion that a square-integrable random variable will concentrate around its mean if its variance is not too large. See these previous notes for more discussion of the concentration of measure phenomenon. One can often obtain stronger concentration of measure than what is provided by Chebyshev’s inequality if one is able to calculate higher moments than the second moment, such as the fourth moment or exponential moments , but we will not pursue this direction in this set of notes.

Clearly the variance is homogeneous of order two, thus

for any and . In particular, the variance is not always additive: the claim fails in particular when is not almost surely zero. However, there is an important substitute for this formula. Given two random variables , the inner product of the corresponding mean zero parts is a complex number known as the covariance:

As are orthogonal to , it is not difficult to obtain the alternate formula

The covariance is then a positive semi-definite inner product on (it basically arises from the Hilbert space structure of the space of mean zero variables), and . From the Cauchy-Schwarz inequality we have

If have non-zero variance (that is, they are not almost surely constant), then the ratio

is then known as the correlation between and , and is a complex number of magnitude at most ; for real-valued that are not almost surely constant, the correlation is instead a real number between and . At one extreme, a correlation of magnitude occurs if and only if is a scalar multiple of . At the other extreme, a correlation of zero is an indication (though not a guarantee) of independence. Recall that two random variables are *independent* if one has

for all (Borel) measurable . In particular, setting , for and integrating using Fubini’s theorem, we conclude that

similarly with replaced by , and similarly for . In particular we have

and thus from (8) we thus see that independent random variables have zero covariance (and zero correlation, when they are not almost surely constant). On the other hand, the converse fails:

Exercise 1Provide an example of two random variables which are not independent, but which have zero correlation or covariance with each other. (There are many ways to produce some examples. One comes from exploiting various systems of orthogonal functions, such as sines and cosines. Another comes from working with random variables taking only a small number of values, such as .

for any finite collection of random variables . These identities combine well with Chebyshev-type inequalities such as (6), (7), and this leads to a very common instance of the second moment method in action. For instance, we can use it to understand the distribution of the number of prime factors of a random number that fall within a given set . Given any set of natural numbers, define the *logarithmic size* to be the quantity

Thus for instance Euler’s theorem asserts that the primes have infinite logarithmic size.

Lemma 2 (Turan-Kubilius inequality, special case)Let be an interval of length at least , and let be an integer drawn uniformly at random from this interval, thusfor all . Let be a finite collection of primes, all of which have size at most . Then the random variable has mean

and variance

In particular,

and from (7) we have

for any .

*Proof:* For any natural number , we have

We now write . From (11) we see that each indicator random variable , has mean and variance ; similarly, for any two distinct , we see from (11), (8) the indicators , have covariance

and the claim now follows from (10).

The exponents of in the error terms here are not optimal; but in practice, we apply this inequality when is much larger than any given power of , so factors such as will be negligible. Informally speaking, the above lemma asserts that a typical number in a large interval will have roughly prime factors in a given finite set of primes, as long as the logarithmic size is large.

If we apply the above lemma to for some large , and equal to the primes up to (say) , we have , and hence

Since , we recover the main result

of Section 5 of Notes 1 (indeed this is essentially the same argument as in that section, dressed up in probabilistic language). In particular, we recover the Hardy-Ramanujan law that a proportion of the natural numbers in have prime factors.

Exercise 3 (Turan-Kubilius inequality, general case)Let be an additive function (which means that whenever are coprime. Show thatwhere

(Hint: one may first want to work with the special case when vanishes whenever so that the second moment method can be profitably applied, and then figure out how to address the contributions of prime powers larger than .)

Exercise 4 (Turan-Kubilius inequality, logarithmic version)Let with , and let be a collection of primes of size less than with . Show that

Exercise 5 (Paley-Zygmund inequality)Let be non-negative with positive mean. Show thatThis inequality can sometimes give slightly sharper results than the Chebyshev inequality when using the second moment method.

Now we give a useful lemma that quantifies a heuristic mentioned in the introduction, namely that if several random variables do not correlate with each other, then it is not possible for any further random variable to correlate with many of them simultaneously. We first state an abstract Hilbert space version.

Lemma 6 (Bessel type inequality, Hilbert space version)If are elements of a Hilbert space , and are positive reals, then

*Proof:* We use the duality method. Namely, we can write the left-hand side of (13) as

for some complex numbers with (just take to be normalised by the left-hand side of (14), or zero if that left-hand side vanishes. By Cauchy-Schwarz, it then suffices to establish the dual inequality

The left-hand side can be written as

Using the arithmetic mean-geometric mean inequality and symmetry, this may be bounded by

Since , the claim follows.

Corollary 7 (Bessel type inequality, probabilistic version)If , and are positive reals, then

*Proof:* By subtracting the mean from each of we may assume that these random variables have mean zero. The claim now follows from Lemma 6.

To get a feel for this inequality, suppose for sake of discussion that and all have unit variance and , but that the are pairwise uncorrelated. Then the right-hand side is equal to , and the left-hand side is the sum of squares of the correlations between and each of the . Any individual correlation is then still permitted to be as large as , but it is not possible for multiple correlations to be this large simultaneously. This is geometrically intuitive if one views the random variables as vectors in a Hilbert space (and correlation as a rough proxy for the angle between such vectors). This lemma also shares many commonalities with the large sieve inequality, discussed in this set of notes.

One basic number-theoretic application of this inequality is the following sampling inequality of Elliott, that lets one approximate a sum of an arithmetic function by its values on multiples of primes :

Exercise 8 (Elliott’s inequality)Let be an interval of length at least . Show that for any function , one has(

Hint:Apply Corollary 7 with , , and , where is the uniform variable from Lemma 2.) Conclude in particular that for every , one hasfor all primes outside of a set of exceptional primes of logarithmic size .

Informally, the point of this inequality is that an arbitrary arithmetic function may exhibit correlation with the indicator function of the multiples of for some primes , but cannot exhibit significant correlation with all of these indicators simultaneously, because these indicators are not very correlated to each other. We note however that this inequality only gains a tiny bit over trivial bounds, because the set of primes up to only has logarithmic size by Mertens’ theorems; thus, any asymptotics that are obtained using this inequality will typically have error terms that only improve upon the trivial bound by factors such as .

Exercise 9 (Elliott’s inequality, logarithmic form)Let with . Show that for any function , one hasand thus for every , one has

for all primes outside of an exceptional set of primes of logarithmic size .

Exercise 10Use Exercise (9) and a duality argument to provide an alternate proof of Exercise 4. (Hint:express the left-hand side of (12) as a correlation between and some suitably -normalised arithmetic function .)

As a quick application of Elliott’s inequality, let us establish a weak version of the prime number theorem:

Proposition 11 (Weak prime number theorem)For any we havewhenever are sufficiently large depending on .

This estimate is weaker than what one can obtain by existing methods, such as Exercise 56 of Notes 1. However in the next section we will refine this argument to recover the full prime number theorem.

*Proof:* Fix , and suppose that are sufficiently large. From Exercise 9 one has

for all primes outside of an exceptional set of logarithmic size . If we restrict attention to primes then one sees from the integral test that one can replace the sum by and only incur an additional error of . If we furthermore restrict to primes larger than , then the contribution of those that are divisible by is also . For not divisible by , one has . Putting all this together, we conclude that

for all primes outside of an exceptional set of logarithmic size . In particular, for large enough this statement is true for at least one such . The claim then follows.

As another application of Elliott’s inequality, we present a criterion for orthogonality between multiplicative functions and other sequences, first discovered by Katai (with related results also introduced earlier by Daboussi and Delange), and rediscovered by Bourgain, Sarnak, and Ziegler:

Proposition 12 (Daboussi-Delange-Katai-Bourgain-Sarnak-Ziegler criterion)Let be a multiplicative function with for all , and let be another bounded function. Suppose that one hasas for any two distinct primes . Then one has

as .

*Proof:* Suppose the claim fails, then there exists (which we can assume to be small) and arbitrarily large such that

By Exercise 8, this implies that

for all primes outside of an exceptional set of logarithmic size . Call such primes “good primes”. In particular, by the pigeonhole principle, and assuming large enough, there exists a dyadic range with which contains good primes.

Fix a good prime in . From (15) we have

We can replace the range by with negligible error. We also have except when is a multiple of , but this latter case only contributes which is also negligible compared to the right-hand side. We conclude that

for every good prime. On the other hand, from Lemma 6 we have

where range over the good primes in . The left-hand side is then , and by hypothesis the right-hand side is for large enough. As and is small, this gives the desired contradiction

Exercise 13 (Daboussi-Delange theorem)Let be irrational, and let be a multiplicative function with for all . Show thatas . If instead is rational, show that there exists be a multiplicative function with for which the statement (16) fails. (Hint: use Dirichlet characters and Plancherel’s theorem for finite abelian groups.)

** — 2. An elementary proof of the prime number theorem — **

Define the Mertens function

As shown in Theorem 58 of Notes 1, the prime number theorem is equivalent to the bound

as . We now give a recent proof of this theorem, due to Redmond McNamara (personal communication), that relies primarily on Elliott’s inequality and the Selberg symmetry formula; it is a relative of the standard elementary proof of this theorem due to Erdös and Selberg. In order to keep the exposition simple, we will not arrange the argument in a fashion that optimises the decay rate (in any event, there are other proofs of the prime number theorem that give significantly stronger bounds).

Firstly we see that Elliott’s inequality gives the following weaker version of (17):

Lemma 14 (Oscillation for Mertens’ function)If and , then we havefor all primes outside of an exceptional set of primes of logarithmic size .

*Proof:* We may assume as the claim is trivial otherwise. From Exercise 8 applied to and , we have

for all outside of an exceptional set of primes of logarithmic size . Since for not divisible by , the right-hand side can be written as

Since outside of an exceptional set of logarithmic size , the claim follows.

Informally, this lemma asserts that for most primes , which morally implies that for most primes . If we can then locate suitable primes with , thus should then lead to , which should then yield the prime number theorem . The manipulations below are intended to make this argument rigorous.

It will be convenient to work with a logarithmically averaged version of this claim.

Corollary 15 (Logarithmically averaged oscillation)If and is sufficiently large depending on , then

*Proof:* For each , we have from the previous lemma that

for all outside of an exceptional set of logarithmic size . We then have

so it suffices by Markov’s inequality to show that

But by Fubini’s theorem, the left-hand side may be bounded by

and the claim follows.

Let be sufficiently small, and let be sufficiently large depending on . Call a prime *good* if the bound (18) holds and *bad* otherwise, thus all primes outside of an exceptional set of bad primes of logarithmic size are good. Now we observe that we can make small as long as we can make two good primes multiply to be close to a third:

*Proof:* By definition of good prime, we have the bounds

We rescale (20) by to conclude that

We can replace the integration range here from to with an error of if is large enough. Also, since , we have . Thus we have

Combining this with (19), (21) and the triangle inequality (writing as a linear combination of , , and ) we conclude that

This is an averaged version of the claim we need. To remove the averaging, we use the identity (see equation (63) of Notes 1) to conclude that

From the triangle inequality one has

and hence by Mertens’ theorem

From the Brun-Titchmarsh inequality (Corollary 61 of Notes 1) we have

and so from the previous estimate and Fubini’s theorem one has

and hence by (22) (using trivial bounds to handle the region outside of )

Since

we conclude (for large enough) that

and the claim follows.

To finish the proof of the prime number theorem, it thus suffices to locate, for sufficiently large, three good primes with . If we already had the prime number theorem, or even the weaker form that every interval of the form contained primes for large enough, then this would be quite easy: pick a large natural number (depending on , but independent of ), so that the primes up to has logarithmic size (so that only of them are bad, as measured by logarithmic size), and let be random numbers and drawn uniformly from (say) . From the prime number theorem, for each , the interval contains primes. In particular, contains primes, but the expected number of bad primes in this interval is . Thus by Markov’s inequality there would be at least a chance (say) of having at least one good prime in ; similarly there is a chance of having a good prime in , and a chance of having a good prime in . Thus (as an application of the probabilistic method), there exist (deterministic) good primes with the required properties.

Of course, using the prime number theorem here to prove the prime number theorem would be circular. However, we can still locate a good triple of primes using the Selberg symmetry formula

as , where is the second von Mangoldt function

see Proposition 60 of Notes 1. We can strip away the contribution of the primes:

Exercise 17Show thatas .

In particular, on evaluating this at and subtracting, we have

whenever is sufficiently large depending on . In particular, for any such , one either has

(or both). Informally, the Selberg symmetry formula shows that the interval contains either a lot of primes, or a lot of semiprimes. The factor of is slightly annoying, so we now remove it. Consider the contribution of those primes to (25) with . This is bounded by

which we can bound crudely using the Chebyshev bound by

which by Mertens theorem is . Thus the contribution of this case can be safely removed from (25). Similarly for those cases when . For the remaining cases we bound . We conclude that for any sufficiently large , either (24) or

In order to find primes with close to , it would be very convenient if we could find a for which (24) and (26) *both* hold. We can’t quite do this directly, but due to the “connected” nature of the set of scales , but we can do the next best thing:

Proposition 18Suppose is sufficiently large depending on . Then there exists with such that

*Proof:* We know that every in obeys at least one of (27), (28). Our task is to produce an adjacent pair of , one of which obeys (27) and the other obeys (28). Suppose for contradiction that no such pair exists, then whenever fails to obey (27), then any adjacent must also fail to do so, and similarly for (28). Thus either (27) will fail to hold for all , or (28) will fail to hold for all such . If (27) fails for all , then on summing we have

which contradicts Mertens’ theorem if is large enough because the left-hand side is . Similarly, if (28) fails for all , then

and again Mertens’ theorem can be used to lower bound the left-hand side by (in fact one can even gain an additional factor of if one works things through carefully) and obtain a contradiction.

The above proposition does indeed provide a triple of primes with . If is sufficiently large depending on and less than (say) , so that , this would give us what we need as long as one of the triples consisted only of good primes. The only way this can fail is if either

for some , or if

for some . In the first case, we can sum to conclude that

and in the second case we have

Since the total set of bad primes up to has logarithmic size , we conclude from the pigeonhole principle (and the divergence of the harmonic series ) that for any depending only on , and any large enough, there exists such that neither of (29) and (30) hold. Indeed the set of obeying (29) has logarithmic size , and similarly for (30). Choosing a that avoids both of these scenarios, we then find a good and good with , so that , and then by Proposition 16 we conclude that for all sufficiently large . Sending to zero, we obtain the prime number theorem.

** — 3. Entropy methods — **

In the previous section we explored the consequences of the second moment method, which applies to square-integrable random variables taking values in the real or complex numbers. Now we explore entropy methods, which now apply to random variables which take a finite number of values (equipped with the discrete sigma-algebra), but whose range need not be numerical in nature. (One could extend entropy methods to slightly larger classes of random variables, such as ones that attain a countable number of values, but for our applications finitely-valued random variables will suffice.)

The fundamental notion here is that of the Shannon entropy of a random variable. If takes values in a finite set , its Shannon entropy (or *entropy* for short) is defined by the formula

where ranges over all the possible values of , and we adopt the convention , so that values that are almost surely not attained by do not influence the entropy. We choose here to use the natural logarithm to normalise our entropy (in which case a unit of entropy is known as a “nat“); in the information theory literature it is also common to use the base two logarithm to measure entropy (in which case a unit of entropy is known as a “bit“, which is equal to nats). However, the precise choice of normalisation will not be important in our discussion.

It is clear that if two random variables have the same probability distribution, then they have the same entropy. Also, the precise choice of range set is not terribly important: if takes values in , and is an injection, then it is clear that and have the same entropy:

This is in sharp contrast to moment-based statistics such as the mean or variance, which can be radically changed by applying some injective transformation to the range values.

Informally, the entropy informally measures how “spread out” or “disordered” the distribution of is, behaving like a logarithm of the size of the “essential support” of such a variable; from an information-theoretic viewpoint, it measures the amount of “information” one learns when one is told the value of . Here are some basic properties of Shannon entropy that help support this intuition:

Exercise 19 (Basic properties of Shannon entropy)Let be a random variable taking values in a finite set .

- (i) Show that , with equality if and only if is almost surely deterministic (that is to say, it is almost surely equal to a constant ).
- (ii) Show that
with equality if and only if is uniformly distributed on . (Hint: use Jensen’s inequality and the convexity of the map on .)

- (iii) (Shannon-McMillan-Breiman theorem) Let be a natural number, and let be independent copies of . As , show that there is a subset of cardinality with the properties that
and

uniformly for all . (The proof of this theorem will require Stirling’s formula, which you may assume here as a black box; see also this previous blog post.) Informally, we thus see a large tuple of independent samples of approximately behaves like a uniform distribution on values.

One can view Shannon entropy as a generalisation of the notion of cardinality of a finite set (or equivalently, cardinality of finite sets can be viewed as a special case of Shannon entropy); see this previous blog post for an elaboration of this point.

The concept of Shannon entropy becomes significantly more powerful when combined with that of conditioning. Recall that a random variable taking values in a range set can be modeled by a measurable map from a probability space to the range . If is an event in of positive probability, we can then *condition* to the event to form a new random variable on the conditioned probability space , where

is the restriction of the -algebra to ,

is the conditional probability measure on , and is the restriction of to . This random variable lives on a different probability space than itself, so it does not make sense to directly combine these variables (thus for instance one cannot form the sum even when both random variables are real or complex valued); however, one can still form the Shannon entropy of the conditioned random variable , which is given by the same formula

Given another random variable taking values in another finite set , we can then define the conditional Shannon entropy to be the expected entropy of the level sets , thus

with the convention that the summand here vanishes when . From the law of total probability we have

for any , and hence by Jensen’s inequality

for any ; summing we obtain the Shannon entropy inequality

Informally, this inequality asserts that the new information content of can be decreased, but not increased, if one is first told some additional information .

This inequality (33) can be rewritten in several ways:

Exercise 20Let , be random variables taking values in finite sets respectively.

- (i) Establish the chain rule
where is the joint random variable . In particular, (33) can be expressed as a subadditivity formula

- (ii) If is a function of , in the sense that for some (deterministic) function , show that .
- (iii) Define the mutual information by the formula
Establish the inequalities

with the first inequality holding with equality if and only if are independent, and the latter inequalities holding if and only if is a function of (or vice versa).

From the above exercise we see that the mutual information is a measure of dependence between and , much as correlation or covariance was in the previous sections. There is however one key difference: whereas a zero correlation or covariance is a consequence but not a guarantee of independence, zero mutual information is *logically equivalent* to independence, and is thus a stronger property. To put it another way, zero correlation or covariance allows one to calculate the average in terms of individual averages of , but zero mutual information is stronger because it allows one to calculate the more general averages in terms of individual averages of , for arbitrary functions taking values into the complex numbers. This increased power of the mutual information statistic will allow us to estimate various averages of interest in analytic number theory in ways that do not seem amenable to second moment methods.

The subadditivity property formula can be conditioned to any event occuring with positive probability (replacing the random variables by their conditioned counterparts ), yielding the inequality

Applying this inequality to the level events of some auxiliary random variable taking values in another finite set , multiplying by , and summing, we conclude the inequality

In other words, the conditional mutual information

between and conditioning on is always non-negative:

One has conditional analogues of the above exercise:

Exercise 21Let , , be random variables taking values in finite sets respectively.

- (i) Establish the conditional chain rule
In particular, (36) is equivalent to the inequality

- (ii) Show that equality holds in (36) if and only if are conditionally independent relative to , which means that
for any , , .

- (iii) Show that , with equality if and only if is almost surely a deterministic function of .
- (iv) Show the data processing inequality
for any functions , , and more generally that

- (v) If is an injective function, show that
However, if is not assumed to be injective, show by means of examples that there is no order relation between the left and right-hand side of (40) (in other words, show that either side may be greater than the other). Thus, increasing or decreasing the amount of information that is known may influence the mutual information between two remaining random variables in either direction.

- (vi) If is a function of , and also a function of (thus for some and ), and a further random variable is a function jointly of (thus for some ), establish the submodularity inequality

We now give a key motivating application of the Shannon entropy inequalities. Suppose one has a sequence of random variables, all taking values in a finite set , which are stationary in the sense that the tuples and have the same distribution for every . In particular we will have

and hence by (39)

If we write , we conclude from (34) that we have the concavity property

In particular we have for any , which on summing and telescoping series (noting that ) gives

and hence we have the entropy monotonicity

In particular, the limit exists. This quantity is known as the Kolmogorov-Sinai entropy of the stationary process ; it is an important statistic in the theory of dynamical systems, and roughly speaking measures the amount of entropy produced by this process as a function of a discrete time vairable . We will not directly need the Kolmogorov-Sinai entropy in our notes, but a variant of the entropy monotonicity formula (41) will be important shortly.

In our application we will be dealing with processes that are only asymptotically stationary rather than stationary. To control this we recall the notion of the total variation distance between two random variables taking values in the same finite space , defined by

There is an essentially equivalent notion of this distance which is also often in use:

Exercise 22If two random variables take values in the same finite space , establish the inequalities

Shannon entropy is continuous in total variation distance as long as we keep the range finite. More quantitatively, we have

Lemma 23If two random variables take values in the same finite space , thenwith the convention that the error term vanishes when .

*Proof:* Set . The claim is trivial when (since then have the same distribution) and when (from (32)), so let us assume , and our task is to show that

If we write , , and , then

By dividing into the cases and we see that

since , it thus suffices to show that

But from Jensen’s inequality (32) one has

since , the claim follows.

In the converse direction, if a random variable has entropy close to the maximum , then one can control the total variation:

Lemma 24 (Special case of Pinsker inequality)If takes values in a finite set , and is a uniformly distributed random variable on , then

Of course, we have , so we may also write the above inequality as

The optimal value of the implied constant here is known to equal , but we will not use this sharp version of the inequality here.

*Proof:* If we write and , and , then we can rewrite the claimed inequality as

Observe that the function is concave, and in fact for all . From this and Taylor expansion with remainder we may write

for some between and . Since is independent of , and , we thus have on summing in

By Cauchy-Schwarz we then have

Since and , the claim follows.

The above lemma does not hold when the comparison variable is not assumed to be uniform; in particular, two non-uniform random variables can have precisely the same entropy but yet have different distributions, so that their total variation distance is positive. There is a more general variant, known as the Pinsker inequality, which we will not use in these notes:

Exercise 25 (Pinsker inequality)If take values in a finite set , define the Kullback-Leibler divergence of relative to by the formula(with the convention that the summand vanishes when vanishes).

- (i) Establish the Gibbs inequality .
- (ii) Establish the Pinsker inequality
In particular, vanishes if and only if have identical distribution. Show that this implies Lemma 24 as a special case.

- (iii) Give an example to show that the Kullback-Liebler divergence need not be symmetric, thus there exist such that .
- (iv) If are random variables taking values in finite sets , and are
independentrandom variables taking values in respectively with each having the same distribution of , show that

In our applications we will need a relative version of Lemma 24:

Corollary 26 (Relative Pinsker inequality)If takes values in a finite set , takes values in a finite set , and is a uniformly distributed random variable on that is independent of , then

*Proof:* From direct calculation we have the identity

As is independent of , is uniformly distributed on . From Lemma 24 we conclude

Inserting this bound and using the Cauchy-Schwarz inequality, we obtain the claim.

Now we are ready to apply the above machinery to give a key inequality that is analogous to Elliott’s inequality. Inequalities of this type first appeared in one of my papers, introducing what I called the “entropy decrement argument”; the following arrangement of the inequality and proof is due to Redmond McNamara (personal communication).

Theorem 27 (Entropy decrement inequality)Let be a random variable taking values in a finite set of integers, which obeys the approximate stationarityfor some . Let be a collection of distinct primes less than some threshold , and let be natural numbers that are also bounded by . Let be a function taking values in a finite set . For , let denote the -valued random variable

and let denote the -valued random variable

Also, let be a random variable drawn uniformly from , independently of . Then

The factor (arising from an invocation of the Chinese remainder theorem in the proof) unfortunately restricts the usefulness of this theorem to the regime in which all the primes involved are of “sub-logarithmic size”, but once one is in that regime, the second term on the right-hand side of (45) tends to be negligible in practice. Informally, this theorem asserts that for most small primes , the random variables and behave as if they are independent of each other.

*Proof:* We can assume , as the claim is trivial for (the all have zero entropy). For , we introduce the -valued random variable

The idea is to exploit some monotonicity properties of the quantity , in analogy with (41). By telescoping series we have

where we extend (44) to the case. From (38) we have

Now we lower bound the summand on the right-hand side. From multiple applications of the conditional chain rule (37) we have

We now use the approximate stationarity of to derive an approximate monotonicity property for . If , then from (39) we have

Write and

Note that is a deterministic function of and vice versa. Thus we can replace by in the above formula, and conclude that

The tuple takes values in a set of cardinality thanks to the Chebyshev bounds. Hence by two applications of Lemma 23, (43) we have

The first term on the right-hand side is . Worsening the error term slightly, we conclude that

and hence

for any . In particular

which by (47), (48) rearranges to

From (46) we conclude that

Meanwhile, from Corollary 26, (39), (38) we have

The probability distribution of is a function on , which by the Chinese remainder theorem we can identify with a cyclic group where . From (43) we see that the value of this distribution at adjacent values of this cyclic group varies by , hence the total variation distance between this random variable and the uniform distribution on is by Chebyshev bounds. By Lemma 23 we then have

and thus

The claim follows.

We now compare this result to Elliott’s inequality. If one tries to address precisely the same question that Elliott’s inequality does – namely, to try to compare a sum with sampled subsums – then the results are quantitatively much weaker:

Corollary 28 (Weak Elliott inequality)Let be an interval of length at least . Let be a function with for all , and let . Then one hasfor all primes outside of an exceptional set of primes of logarithmic size .

Comparing this with Exercise 8 we see that we cover a much smaller range of primes ; also the size of the exceptional set is slightly worse. This version of Elliot’s inequality is however still strong enough to recover a proof of the prime number theorem as in the previous section.

*Proof:* We can assume that is small, as the claim is trivial for comparable to . We can also assume that

since the claim is also trivial otherwise (just make all primes up to exceptional, then use Mertens’ theorem). As a consequence of this, any quantity involving in the denominator will end up being completely negligible in practice. We can also restrict attention to primes less than (say) , since the remaining primes between and have logarithmic size .

By rounding the real and imaginary parts of to the nearest multiple of , we may assume that takes values in some finite set of complex numbers of size with cardinality . Let be drawn uniformly at random from . Then (43) holds with , and from Theorem 27 with and (which makes the second term of the right-hand side of (45) negligible) we have

where are the primes up to , arranged in increasing order. By Markov’s inequality, we thus have

for outside of a set of primes of logarithmic size .

Let be as above. Now let be the function

that is to say picks out the unique component of the tuple in which is divisible by . This function is bounded by , and then by (42) we have

The left-hand side is equal to

which on switching the summations and using the large nature of can be rewritten as

Meanwhile, the left-hand side is equal to

which again by switching the summations becomes

The claim follows.

In the above argument we applied (42) with a very specific choice of function . The power of Theorem 27 lies in the ability to select many other such functions , leading to estimates that do not seem to be obtainable purely from the second moment method. In particular we have the following generalisation of the previous estimate:

Proposition 29 (Weak Elliott inequality for multiple correlations)Let be an interval of length at least . Let be a function with for all , and let . Let be integers. Then one hasfor all primes outside of an exceptional set of primes of logarithmic size .

*Proof:* We allow all implied constants to depend on . As before we can assume that is sufficiently small (depending on ), that takes values in a set of bounded complex numbers of cardinality , and that is large in the sense of (49), and restrict attention to primes up to . By shifting the and using the large nature of we can assume that the are all non-negative, taking values in for some . We now apply Theorem 27 with and conclude as before that

for outside of a set of primes of logarithmic size .

Let be as above. Let be the function

This function is still bounded by , so by (42) as before we have

The left-hand side is equal to

which on switching the summations and using the large nature of can be rewritten as

Meanwhile, the left-hand side is equal to

which again by switching the summations becomes

The claim follows.

There is a logarithmically averaged version of the above proposition:

Exercise 30 (Weak Elliott inequality for logarithmically averaged multiple correlations)Let with , let be a function bounded in magnitude by , let , and let be integers. Show thatfor all primes outside of an exceptional set of primes of logarithmic size .

When one specialises to multiplicative functions, this lets us dilate shifts in multiple correlations by primes:

Exercise 31Let with , let be a multiplicative function bounded in magnitude by , let , and let be nonnegative integers. Show thatfor all primes outside of an exceptional set of primes of logarithmic size .

For instance, setting to be the Möbius function, , , and (say), we see that

for all primes outside of an exceptional set of primes of logarithmic size . In particular, for large enough, one can obtain bounds of the form

for various moderately large sets of primes . It turns out that these double sums on the right-hand side can be estimated by methods which we will cover in later series of notes. Among other things, this allows us to establish estimates such as

as , which to date have only been established using these entropy methods (in conjunction with the methods discussed in later notes). This is progress towards an open problem in analytic number theory known as *Chowla’s conjecture*, which we will also discuss in later notes.

Conjecture 1 (Collatz conjecture)One has for all .

Establishing the conjecture for all remains out of reach of current techniques (for instance, as discussed in the previous blog post, it is basically at least as difficult as Baker’s theorem, all known proofs of which are quite difficult). However, the situation is more promising if one is willing to settle for results which only hold for “most” in some sense. For instance, it is a result of Krasikov and Lagarias that

for all sufficiently large . In another direction, it was shown by Terras that for almost all (in the sense of natural density), one has . This was then improved by Allouche to , and extended later by Korec to cover all . In this paper we obtain the following further improvement (at the cost of weakening natural density to logarithmic density):

Theorem 2Let be any function with . Then we have for almost all (in the sense of logarithmic density).

Thus for instance one has for almost all (in the sense of logarithmic density).

The difficulty here is one usually only expects to establish “local-in-time” results that control the evolution for times that only get as large as a small multiple of ; the aforementioned results of Terras, Allouche, and Korec, for instance, are of this type. However, to get all the way down to one needs something more like an “(almost) global-in-time” result, where the evolution remains under control for so long that the orbit has nearly reached the bounded state .

However, as observed by Bourgain in the context of nonlinear Schrödinger equations, one can iterate “almost sure local wellposedness” type results (which give local control for almost all initial data from a given distribution) into “almost sure (almost) global wellposedness” type results if one is fortunate enough to draw one’s data from an *invariant measure* for the dynamics. To illustrate the idea, let us take Korec’s aforementioned result that if one picks at random an integer from a large interval , then in most cases, the orbit of will eventually move into the interval . Similarly, if one picks an integer at random from , then in most cases, the orbit of will eventually move into . It is then tempting to concatenate the two statements and conclude that for most in , the orbit will eventually move . Unfortunately, this argument does not quite work, because by the time the orbit from a randomly drawn reaches , the distribution of the final value is unlikely to be close to being uniformly distributed on , and in particular could potentially concentrate almost entirely in the exceptional set of that do not make it into . The point here is the uniform measure on is not transported by Collatz dynamics to anything resembling the uniform measure on .

So, one now needs to locate a measure which has better invariance properties under the Collatz dynamics. It turns out to be technically convenient to work with a standard acceleration of the Collatz map known as the *Syracuse map* , defined on the odd numbers by setting , where is the largest power of that divides . (The advantage of using the Syracuse map over the Collatz map is that it performs precisely one multiplication of at each iteration step, which makes the map better behaved when performing “-adic” analysis.)

When viewed -adically, we soon see that iterations of the Syracuse map become somewhat irregular. Most obviously, is never divisible by . A little less obviously, is twice as likely to equal mod as it is to equal mod . This is because for a randomly chosen odd , the number of times that divides can be seen to have a geometric distribution of mean – it equals any given value with probability . Such a geometric random variable is twice as likely to be odd as to be even, which is what gives the above irregularity. There are similar irregularities modulo higher powers of . For instance, one can compute that for large random odd , will take the residue classes with probabilities

respectively. More generally, for any , will be distributed according to the law of a random variable on that we call a *Syracuse random variable*, and can be described explicitly as

where are iid copies of a geometric random variable of mean .

In view of this, any proposed “invariant” (or approximately invariant) measure (or family of measures) for the Syracuse dynamics should take this -adic irregularity of distribution into account. It turns out that one can use the Syracuse random variables to construct such a measure, but only if these random variables stabilise in the limit in a certain total variation sense. More precisely, in the paper we establish the estimate

for any and any . This type of stabilisation is plausible from entropy heuristics – the tuple of geometric random variables that generates has Shannon entropy , which is significantly larger than the total entropy of the uniform distribution on , so we expect a lot of “mixing” and “collision” to occur when converting the tuple to ; these heuristics can be supported by numerics (which I was able to work out up to about before running into memory and CPU issues), but it turns out to be surprisingly delicate to make this precise.

A first hint of how to proceed comes from the elementary number theory observation (easily proven by induction) that the rational numbers

are all distinct as vary over tuples in . Unfortunately, the process of reducing mod creates a lot of collisions (as must happen from the pigeonhole principle); however, by a simple “Lefschetz principle” type argument one can at least show that the reductions

are mostly distinct for “typical” (as drawn using the geometric distribution) as long as is a bit smaller than (basically because the rational number appearing in (3) then typically takes a form like with an integer between and ). This analysis of the component (3) of (1) is already enough to get quite a bit of spreading on (roughly speaking, when the argument is optimised, it shows that this random variable cannot concentrate in any subset of of density less than for some large absolute constant ). To get from this to a stabilisation property (2) we have to exploit the mixing effects of the remaining portion of (1) that does not come from (3). After some standard Fourier-analytic manipulations, matters then boil down to obtaining non-trivial decay of the characteristic function of , and more precisely in showing that

for any and any that is not divisible by .

If the random variable (1) was the sum of independent terms, one could express this characteristic function as something like a Riesz product, which would be straightforward to estimate well. Unfortunately, the terms in (1) are loosely coupled together, and so the characteristic factor does not immediately factor into a Riesz product. However, if one groups adjacent terms in (1) together, one can rewrite it (assuming is even for sake of discussion) as

where . The point here is that after conditioning on the to be fixed, the random variables remain independent (though the distribution of each depends on the value that we conditioned to), and so the above expression is a *conditional* sum of independent random variables. This lets one express the characeteristic function of (1) as an *averaged* Riesz product. One can use this to establish the bound (4) as long as one can show that the expression

is not close to an integer for a moderately large number (, to be precise) of indices . (Actually, for technical reasons we have to also restrict to those for which , but let us ignore this detail here.) To put it another way, if we let denote the set of pairs for which

we have to show that (with overwhelming probability) the random walk

(which we view as a two-dimensional renewal process) contains at least a few points lying outside of .

A little bit of elementary number theory and combinatorics allows one to describe the set as the union of “triangles” with a certain non-zero separation between them. If the triangles were all fairly small, then one expects the renewal process to visit at least one point outside of after passing through any given such triangle, and it then becomes relatively easy to then show that the renewal process usually has the required number of points outside of . The most difficult case is when the renewal process passes through a particularly large triangle in . However, it turns out that large triangles enjoy particularly good separation properties, and in particular afer passing through a large triangle one is likely to only encounter nothing but small triangles for a while. After making these heuristics more precise, one is finally able to get enough points on the renewal process outside of that one can finish the proof of (4), and thus Theorem 2.

]]>- Elementary multiplicative number theory
- Complex-analytic multiplicative number theory
- The entropy decrement argument
- Bounds for exponential sums
- Zero density theorems
- Halasz’s theorem and the Matomaki-Radziwill theorem
- The circle method
- (If time permits) Chowla’s conjecture and the Erdos discrepancy problem

Lecture notes for topics 3, 6, and 8 will be forthcoming.

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Conjecture 1 (Cramér conjecture)If is a large number, then the largest prime gap in is of size . (Granville refines this conjecture to , where . Here we use the asymptotic notation for , for , for , and for .)

Conjecture 2 (Hardy-Littlewood conjecture)If are fixed distinct integers, then the number of numbers with all prime is as , where the singular series is defined by the formula

(One can view these conjectures as modern versions of two of the classical Landau problems, namely Legendre’s conjecture and the twin prime conjecture respectively.)

A well known connection between the Hardy-Littlewood conjecture and prime gaps was made by Gallagher. Among other things, Gallagher showed that if the Hardy-Littlewood conjecture was true, then the prime gaps with were asymptotically distributed according to an exponential distribution of mean , in the sense that

as for any fixed . Roughly speaking, the way this is established is by using the Hardy-Littlewood conjecture to control the mean values of for fixed , where ranges over the primes in . The relevance of these quantities arises from the Bonferroni inequalities (or “Brun pure sieve“), which can be formulated as the assertion that

when is even and

when is odd, for any natural number ; setting and taking means, one then gets upper and lower bounds for the probability that the interval is free of primes. The most difficult step is to control the mean values of the singular series as ranges over -tuples in a fixed interval such as .

Heuristically, if one extrapolates the asymptotic (1) to the regime , one is then led to Cramér’s conjecture, since the right-hand side of (1) falls below when is significantly larger than . However, this is not a rigorous derivation of Cramér’s conjecture from the Hardy-Littlewood conjecture, since Gallagher’s computations only establish (1) for *fixed* choices of , which is only enough to establish the far weaker bound , which was already known (see this previous paper for a discussion of the best known unconditional lower bounds on ). An inspection of the argument shows that if one wished to extend (1) to parameter choices that were allowed to grow with , then one would need as input a stronger version of the Hardy-Littlewood conjecture in which the length of the tuple , as well as the magnitudes of the shifts , were also allowed to grow with . Our initial objective in this project was then to quantify exactly what strengthening of the Hardy-Littlewood conjecture would be needed to rigorously imply Cramer’s conjecture. The precise results are technical, but roughly we show results of the following form:

Theorem 3 (Large gaps from Hardy-Littlewood, rough statement)

- If the Hardy-Littlewood conjecture is uniformly true for -tuples of length , and with shifts of size , with a power savings in the error term, then .
- If the Hardy-Littlewood conjecture is “true on average” for -tuples of length and shifts of size for all , with a power savings in the error term, then .

In particular, we can recover Cramer’s conjecture given a sufficiently powerful version of the Hardy-Littlewood conjecture “on the average”.

Our proof of this theorem proceeds more or less along the same lines as Gallagher’s calculation, but now with allowed to grow slowly with . Again, the main difficulty is to accurately estimate average values of the singular series . Here we found it useful to switch to a probabilistic interpretation of this series. For technical reasons it is convenient to work with a truncated, unnormalised version

of the singular series, for a suitable cutoff ; it turns out that when studying prime tuples of size , the most convenient cutoff is the “Pólya magic cutoff“, defined as the largest prime for which

(this is well defined for ); by Mertens’ theorem, we have . One can interpret probabilistically as

where is the randomly sifted set of integers formed by removing one residue class uniformly at random for each prime . The Hardy-Littlewood conjecture can be viewed as an assertion that the primes behave in some approximate statistical sense like the random sifted set , and one can prove the above theorem by using the Bonferroni inequalities both for the primes and for the random sifted set, and comparing the two (using an even for the sifted set and an odd for the primes in order to be able to combine the two together to get a useful bound).

The proof of Theorem 3 ended up not using any properties of the set of primes other than that this set obeyed some form of the Hardy-Littlewood conjectures; the theorem remains true (with suitable notational changes) if this set were replaced by any other set. In order to convince ourselves that our theorem was not vacuous due to our version of the Hardy-Littlewood conjecture being too strong to be true, we then started exploring the question of coming up with random models of which obeyed various versions of the Hardy-Littlewood and Cramér conjectures.

This line of inquiry was started by Cramér, who introduced what we now call the *Cramér random model* of the primes, in which each natural number is selected for membership in with an independent probability of . This model matches the primes well in some respects; for instance, it almost surely obeys the “Riemann hypothesis”

and Cramér also showed that the largest gap was almost surely . On the other hand, it does not obey the Hardy-Littlewood conjecture; more precisely, it obeys a simplified variant of that conjecture in which the singular series is absent.

Granville proposed a refinement to Cramér’s random model in which one first sieves out (in each dyadic interval ) all residue classes for for a certain threshold , and then places each surviving natural number in with an independent probability . One can verify that this model obeys the Hardy-Littlewood conjectures, and Granville showed that the largest gap in this model was almost surely , leading to his conjecture that this bound also was true for the primes. (Interestingly, this conjecture is not yet borne out by numerics; calculations of prime gaps up to , for instance, have shown that never exceeds in this range. This is not necessarily a conflict, however; Granville’s analysis relies on inspecting gaps in an extremely sparse region of natural numbers that are more devoid of primes than average, and this region is not well explored by existing numerics. See this previous blog post for more discussion of Granville’s argument.)

However, Granville’s model does not produce a power savings in the error term of the Hardy-Littlewood conjectures, mostly due to the need to truncate the singular series at the logarithmic cutoff . After some experimentation, we were able to produce a tractable random model for the primes which obeyed the Hardy-Littlewood conjectures with power savings, and which reproduced Granville’s gap prediction of (we also get an upper bound of for both models, though we expect the lower bound to be closer to the truth); to us, this strengthens the case for Granville’s version of Cramér’s conjecture. The model can be described as follows. We select one residue class uniformly at random for each prime , and as before we let be the sifted set of integers formed by deleting the residue classes with . We then set

with Pólya’s magic cutoff (this is the cutoff that gives a density consistent with the prime number theorem or the Riemann hypothesis). As stated above, we are able to show that almost surely one has

and that the Hardy-Littlewood conjectures hold with power savings for up to for any fixed and for shifts of size . This is unfortunately a tiny bit weaker than what Theorem 3 requires (which more or less corresponds to the endpoint ), although there is a variant of Theorem 3 that can use this input to produce a lower bound on gaps in the model (but it is weaker than the one in (3)). In fact we prove a more precise almost sure asymptotic formula for that involves the optimal bounds for the *linear sieve* (or *interval sieve*), in which one deletes one residue class modulo from an interval for all primes up to a given threshold. The lower bound in (3) relates to the case of deleting the residue classes from ; the upper bound comes from the delicate analysis of the linear sieve by Iwaniec. Improving on either of the two bounds looks to be quite a difficult problem.

The probabilistic analysis of is somewhat more complicated than of or as there is now non-trivial coupling between the events as varies, although moment methods such as the second moment method are still viable and allow one to verify the Hardy-Littlewood conjectures by a lengthy but fairly straightforward calculation. To analyse large gaps, one has to understand the statistical behaviour of a random linear sieve in which one starts with an interval and randomly deletes a residue class for each prime up to a given threshold. For very small this is handled by the deterministic theory of the linear sieve as discussed above. For medium sized , it turns out that there is good concentration of measure thanks to tools such as Bennett’s inequality or Azuma’s inequality, as one can view the sieving process as a martingale or (approximately) as a sum of independent random variables. For larger primes , in which only a small number of survivors are expected to be sieved out by each residue class, a direct combinatorial calculation of all possible outcomes (involving the random graph that connects interval elements to primes if falls in the random residue class ) turns out to give the best results.

]]>that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. There are several such results that assert that if a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular; taken in the contrapositive, they assert that if a solution blows up at a certain finite time , then certain norms of the solution must also go to infinity. Here are some examples (not an exhaustive list) of such blowup criteria:

- (Leray blowup criterion, 1934) If blows up at a finite time , and , then for an absolute constant .
- (Prodi–Serrin–Ladyzhenskaya blowup criterion, 1959-1967) If blows up at a finite time , and , then , where .
- (Beale-Kato-Majda blowup criterion, 1984) If blows up at a finite time , then , where is the vorticity.
- (Kato blowup criterion, 1984) If blows up at a finite time , then for some absolute constant .
- (Escauriaza-Seregin-Sverak blowup criterion, 2003) If blows up at a finite time , then .
- (Seregin blowup criterion, 2012) If blows up at a finite time , then .
- (Phuc blowup criterion, 2015) If blows up at a finite time , then for any .
- (Gallagher-Koch-Planchon blowup criterion, 2016) If blows up at a finite time , then for any .
- (Albritton blowup criterion, 2016) If blows up at a finite time , then for any .

My current paper is most closely related to the Escauriaza-Seregin-Sverak blowup criterion, which was the first to show a critical (i.e., scale-invariant, or dimensionless) spatial norm, namely , had to become large. This result now has many proofs; for instance, many of the subsequent blowup criterion results imply the Escauriaza-Seregin-Sverak one as a special case, and there are also additional proofs by Gallagher-Koch-Planchon (building on ideas of Kenig-Koch), and by Dong-Du. However, all of these proofs rely on some form of a compactness argument: given a finite time blowup, one extracts some suitable family of rescaled solutions that converges in some weak sense to a limiting solution that has some additional good properties (such as almost periodicity modulo symmetries), which one can then rule out using additional qualitative tools, such as unique continuation and backwards uniqueness theorems for parabolic heat equations. In particular, all known proofs use some version of the backwards uniqueness theorem of Escauriaza, Seregin, and Sverak. Because of this reliance on compactness, the existing proofs of the Escauriaza-Seregin-Sverak blowup criterion are qualitative, in that they do not provide any quantitative information on how fast the norm will go to infinity (along a subsequence of times).

On the other hand, it is a general principle that qualitative arguments established using compactness methods ought to have quantitative analogues that replace the use of compactness by more complicated substitutes that give effective bounds; see for instance these previous blog posts for more discussion. I therefore was interested in trying to obtain a quantitative version of this blowup criterion that gave reasonably good effective bounds (in particular, my objective was to avoid truly enormous bounds such as tower-exponential or Ackermann function bounds, which often arise if one “naively” tries to make a compactness argument effective). In particular, I obtained the following triple-exponential quantitative regularity bounds:

Theorem 1If is a classical solution to Navier-Stokes on with

and

for and .

As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to

for some absolute constant , which to my knowledge is the first (*very* slightly) supercritical blowup criterion for Navier-Stokes in the literature.

The proof uses many of the same quantitative inputs as previous arguments, most notably the Carleman inequalities used to establish unique continuation and backwards uniqueness theorems for backwards heat equations, but also some additional techniques that make the quantitative bounds more efficient. The proof focuses initially on points of concentration of the solution, which we define as points where there is a frequency for which one has the bound

for a large absolute constant , where is a Littlewood-Paley projection to frequencies . (This can be compared with the upper bound of for the quantity on the left-hand side that follows from (1).) The factor of normalises the left-hand side of (2) to be dimensionless (i.e., critical). The main task is to show that the dimensionless quantity cannot get too large; in particular, we end up establishing a bound of the form

from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.

The strategy is to show that any concentration such as (2) when is large must force a significant component of the norm of to also show up at many other locations than , which eventually contradicts (1) if one can produce enough such regions of non-trivial norm. (This can be viewed as a quantitative variant of the “rigidity” theorems in some of the previous proofs of the Escauriaza-Seregin-Sverak theorem that rule out solutions that exhibit too much “compactness” or “almost periodicity” in the topology.) The chain of causality that leads from a concentration (2) at to significant norm at other regions of the time slice is somewhat involved (though simpler than the much more convoluted schemes I initially envisaged for this argument):

- Firstly, by using Duhamel’s formula, one can show that a concentration (2) can only occur (with large) if there was also a preceding concentration
at some slightly previous point in spacetime, with also close to (more precisely, we have , , and ). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound in the conclusion (3) is precisely the same as the lower bound in (2), so that this backwards propagation of concentration can be iterated.

- Iterating the previous step, one can find a sequence of concentration points
with the propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the increase geometrically with time, the are comparable (up to polynomial factors in ) to the natural frequency scale , and one has . Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the slightly, one can also place the times in intervals (of length comparable to a small multiple of ) in which the solution is quite regular (in particular, enjoy good bounds on ).

- The concentration (4) can be used to establish a lower bound for the norm of the vorticity near . As is well known, the vorticity obeys the vorticity equation
In the epoch of regularity , the coefficients of this equation obey good bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower bounds on the vorticity (and its first derivative) in annuli of the form for various radii , although the lower bounds decay at a gaussian rate with .

- Meanwhile, using an energy pigeonholing argument of Bourgain (which, in this Navier-Stokes context, is actually an enstrophy pigeonholing argument), one can locate some annuli where (a slightly normalised form of) the entrosphy is small at time ; using a version of the localised enstrophy estimates from a previous paper of mine, one can then propagate this sort of control forward in time, obtaining an “annulus of regularity” of the form in which one has good estimates; in particular, one has type bounds on in this cylindrical annulus.
- By intersecting the previous epoch of regularity with the above annulus of regularity, we have some lower bounds on the norm of the vorticity (and its first derivative) in the annulus of regularity. Using a Carleman estimate first introduced by Escauriaza, Seregin, and Sverak, as well as a second application of the Carleman estimate used previously, one can then propagate this lower bound back up to time , establishing a lower bound for the vorticity on the spatial annulus . By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the norm of the velocity ; crucially, this lower bound is uniform in .
- If is very large (triple exponential in !), one can then find enough scales with disjoint annuli that the total lower bound on the norm of provided by the above arguments is inconsistent with (1), thus establishing the claim.

The chain of causality is summarised in the following image:

It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the upper bound costs the third.

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Theorem 1Let be an Hermitian matrix, with eigenvalues . Let be a unit eigenvector corresponding to the eigenvalue , and let be the component of . Thenwhere is the Hermitian matrix formed by deleting the row and column from .

For instance, if we have

for some real number , -dimensional vector , and Hermitian matrix , then we have

assuming that the denominator is non-zero.

Once one is aware of the identity, it is not so difficult to prove it; we give two proofs, each about half a page long, one of which is based on a variant of the Cauchy-Binet formula, and the other based on properties of the adjugate matrix. But perhaps it is surprising that such a formula exists at all; one does not normally expect to learn much information about eigenvectors purely from knowledge of eigenvalues. In the random matrix theory literature, for instance in this paper of Erdos, Schlein, and Yau, or this later paper of Van Vu and myself, a related identity has been used, namely

but it is not immediately obvious that one can derive the former identity from the latter. (I do so below the fold; we ended up not putting this proof in the note as it was longer than the two other proofs we found. I also give two other proofs below the fold, one from a more geometric perspective and one proceeding via Cramer’s rule.) It was certainly something of a surprise to me that there is no explicit appearance of the components of in the formula (1) (though they do indirectly appear through their effect on the eigenvalues ; for instance from taking traces one sees that ).

One can get some feeling of the identity (1) by considering some special cases. Suppose for instance that is a diagonal matrix with all distinct entries. The upper left entry of is one of the eigenvalues of . If it is equal to , then the eigenvalues of are the other eigenvalues of , and now the left and right-hand sides of (1) are equal to . At the other extreme, if is equal to a different eigenvalue of , then now appears as an eigenvalue of , and both sides of (1) now vanish. More generally, if we order the eigenvalues and , then the Cauchy interlacing inequalities tell us that

for , and

for , so that the right-hand side of (1) lies between and , which is of course consistent with (1) as is a unit vector. Thus the identity relates the coefficient sizes of an eigenvector with the extent to which the Cauchy interlacing inequalities are sharp.

** — 1. Relating the two identities — **

We now show how (1) can be deduced from (2). By a limiting argument, it suffices to prove (1) in the case when is not an eigenvalue of . Without loss of generality we may take . By subtracting the matrix from (and from , thus shifting all the eigenvalues down by , we may also assume without loss of generality that . So now we wish to show that

The right-hand side is just . If one differentiates the characteristic polynomial

at , one sees that

Finally, (2) can be rewritten as

so our task is now to show that

By Schur complement, we have

Since is an eigenvalue of , but not of (by hypothesis), the factor vanishes when . If we then differentiate (4) in and set we obtain (3) as desired.

** — 2. A geometric proof — **

Here is a more geometric way to think about the identity. One can view as a linear operator on (mapping to for any vector ); it then also acts on all the exterior powers by mapping to for all vectors . In particular, if one evaluates on the basis of induced by the orthogonal eigenbasis , we see that the action of on is rank one, with

for any , where is the inner product on induced by the standard inner product on . If we now apply this to the -form , we have , while is equal to plus some terms orthogonal to . Since , Theorem 1 follows.

** — 3. A proof using Cramer’s rule — **

By a limiting argument we can assume that all the eigenvalues of are simple. From the spectral theorem we can compute the resolvent for as

Extracting the component of both sides and using Cramer’s rule, we conclude that

or in terms of eigenvalues

Both sides are rational functions with a simple pole at the eigenvalues . Extracting the residue at we conclude that

and Theorem 1 follows. (Note that this approach also gives a formula for for , although the formula becomes messier when because the relevant minor of is no longer a scalar multiple of the identity .)

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Theorem 1 (Multilinear Kakeya estimate)Let be a radius. For each , let denote a finite family of infinite tubes in of radius . Assume the following axiom:

- (i) (Transversality) whenever is oriented in the direction of a unit vector for , we have
for some , where we use the usual Euclidean norm on the wedge product .

where are the usual Lebesgue norms with respect to Lebesgue measure, denotes the indicator function of , and denotes the cardinality of .

The original proof of this proceeded using a heat flow monotonicity method, which in my previous post I reinterpreted using a “virtual integration” concept on a fractional Cartesian product space. It turns out that this machinery is somewhat flexible, and can be used to establish some other estimates of this type. The first result of this paper is to extend the above theorem to the curved setting, in which one localises to a ball of radius (and sets to be small), but allows the tubes to be curved in a fashion. If one runs the heat flow monotonicity argument, one now picks up some additional error terms arising from the curvature, but as the spatial scale approaches zero, the tubes become increasingly linear, and as such the error terms end up being an integrable multiple of the main term, at which point one can conclude by Gronwall’s inequality (actually for technical reasons we use a bootstrap argument instead of Gronwall). A key point in this approach is that one obtains optimal bounds (not losing factors of or ), so long as one stays away from the endpoint case (which does not seem to be easily treatable by the heat flow methods). Previously, the paper of Bennett, Carbery, and myself was able to use an induction on scale argument to obtain a curved multilinear Kakeya estimate losing a factor of (after optimising the argument); later arguments of Bourgain-Guth and Carbery-Valdimarsson, based on algebraic topology methods, could also obtain a curved multilinear Kakeya estimate without such losses, but only in the algebraic case when the tubes were neighbourhoods of algebraic curves of bounded degree.

Perhaps more interestingly, we are also able to extend the heat flow monotonicity method to apply directly to the multilinear restriction problem, giving the following global multilinear restriction estimate:

Theorem 2 (Multilinear restriction theorem)Let be an exponent, and let be a parameter. Let be a sufficiently large natural number, depending only on . For , let be an open subset of , and let be a smooth function obeying the following axioms:for any , , extended by zero outside of , and denotes the extension operator

Local versions of such estimate, in which is replaced with for some , and one accepts a loss of the form , were already established by Bennett, Carbery, and myself using an induction on scale argument. In a later paper of Bourgain-Guth these losses were removed by “epsilon removal lemmas” to recover Theorme 2, but only in the case when all the hypersurfaces involved had curvatures bounded away from zero.

There are two main new ingredients in the proof of Theorem 2. The first is to replace the usual induction on scales scheme to establish multilinear restriction by a “ball inflation” induction on scales scheme that more closely resembles the proof of decoupling theorems. In particular, we actually prove the more general family of estimates

where denotes the local energies

(actually for technical reasons it is more convenient to use a smoother weight than the strict cutoff to the disk ). With logarithmic losses, it is not difficult to establish this estimate by an upward induction on . To avoid such losses we use the heat flow monotonicity method. Here we run into the issue that the extension operators are complex-valued rather than non-negative, and thus would not be expected to obey many good montonicity properties. However, the local energies can be expressed in terms of the magnitude squared of what is essentially the Gabor transform of , and these are non-negative; furthermore, the dispersion relation associated to the extension operators implies that these Gabor transforms propagate along tubes, so that the situation becomes quite similar (up to several additional lower order error terms) to that in the multilinear Kakeya problem. (This can be viewed as a continuous version of the usual wave packet decomposition method used to relate restriction and Kakeya problems, which when combined with the heat flow monotonicity method allows for one to use a continuous version of induction on scales methods that do not concede any logarithmic factors.)

Finally, one can combine the curved multilinear Kakeya result with the multilinear restriction result to obtain estimates for multilinear oscillatory integrals away from the endpoint. Again, this sort of implication was already established in the previous paper of Bennett, Carbery, and myself, but the arguments there had some epsilon losses in the exponents; here we were able to run the argument more carefully and avoid these losses.

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Theorem 1 (Lower bound on maximum degree of induced subgraphs of hypercube)Let be a set of at least vertices in . Then there is a vertex in that is adjacent (in ) to at least other vertices in .

The bound (or more precisely, ) is completely sharp, as shown by Chung, Furedi, Graham, and Seymour; we describe this example below the fold. When combined with earlier reductions of Gotsman-Linial and Nisan-Szegedy; we give these below the fold also.

Let be the adjacency matrix of (where we index the rows and columns directly by the vertices in , rather than selecting some enumeration ), thus when for some , and otherwise. The above theorem then asserts that if is a set of at least vertices, then the minor of has a row (or column) that contains at least non-zero entries.

The key step to prove this theorem is the construction of rather curious variant of the adjacency matrix :

Proposition 2There exists a matrix which is entrywise dominated by in the sense that

Assuming this proposition, the proof of Theorem 1 can now be quickly concluded. If we view as a linear operator on the -dimensional space of functions of , then by hypothesis this space has a -dimensional subspace on which acts by multiplication by . If is a set of at least vertices in , then the space of functions on has codimension at most in , and hence intersects non-trivially. Thus the minor of also has as an eigenvalue (this can also be derived from the Cauchy interlacing inequalities), and in particular this minor has operator norm at least . By Schur’s test, this implies that one of the rows or columns of this matrix has absolute values summing to at least , giving the claim.

Remark 3The argument actually gives a strengthening of Theorem 1: there exists a vertex of with the property that for every natural number , there are at least paths of length in the restriction of to that start from . Indeed, if we let be an eigenfunction of on , and let be a vertex in that maximises the value of , then for any we have that the component of is equal to ; on the other hand, by the triangle inequality, this component is at most times the number of length paths in starting from , giving the claim.

This argument can be viewed as an instance of a more general “interlacing method” to try to control the behaviour of a graph on all large subsets by first generating a matrix on with very good spectral properties, which are then partially inherited by the minor of by interlacing inequalities. In previous literature using this method (see e.g., this survey of Haemers, or this paper of Wilson), either the original adjacency matrix , or some non-negatively weighted version of that matrix, was used as the controlling matrix ; the novelty here is the use of signed controlling matrices. It will be interesting to see what further variants and applications of this method emerge in the near future. (Thanks to Anurag Bishoi in the comments for these references.)

The “magic” step in the above argument is constructing . In Huang’s paper, is constructed recursively in the dimension in a rather simple but mysterious fashion. Very recently, Roman Karasev gave an interpretation of this matrix in terms of the exterior algebra on . In this post I would like to give an alternate interpretation in terms of the operation of *twisted convolution*, which originated in the theory of the Heisenberg group in quantum mechanics.

Firstly note that the original adjacency matrix , when viewed as a linear operator on , is a convolution operator

where

is the counting measure on the standard basis , and denotes the ordinary convolution operation

As is well known, this operation is commutative and associative. Thus for instance the square of the adjacency operator is also a convolution operator

where the convolution kernel is moderately complicated:

The factor in this expansion comes from combining the two terms and , which both evaluate to .

More generally, given any bilinear form , one can define the *twisted convolution*

of two functions . This operation is no longer commutative (unless is symmetric). However, it remains associative; indeed, one can easily compute that

In particular, if we define the twisted convolution operator

then the square is also a twisted convolution operator

and the twisted convolution kernel can be computed as

For general bilinear forms , this twisted convolution is just as messy as is. But if we take the specific bilinear form

then for and for , and the above twisted convolution simplifies to

and now is very simple:

Thus the only eigenvalues of are and . The matrix is entrywise dominated by in the sense of (1), and in particular has trace zero; thus the and eigenvalues must occur with equal multiplicity, so in particular the eigenvalue occurs with multiplicity since the matrix has dimensions . This establishes Proposition 2.

Remark 4Twisted convolution is actually just a component of ordinary convolution, but not on the original group ; instead it relates to convolution on a Heisenberg group extension of this group. More specifically, define the Heisenberg group to be the set of pairs with group lawand inverse operation

(one can dispense with the negative signs here if desired, since we are in characteristic two). Convolution on is defined in the usual manner: one has

for any . Now if is a function on the original group , we can define the lift by the formula

and then by chasing all the definitions one soon verifies that

for any , thus relating twisted convolution to Heisenberg group convolution .

Remark 5With the twisting by the specific bilinear form given by (2), convolution by and now anticommute rather than commute. This makes the twisted convolution algebra isomorphic to a Clifford algebra (the real or complex algebra generated by formal generators subject to the relations for ) rather than the commutative algebra more familiar to abelian Fourier analysis. This connection to Clifford algebra (also observed independently by Tom Mrowka and by Daniel Matthews) may be linked to the exterior algebra interpretation of the argument in the recent preprint of Karasev mentioned above.

Remark 6One could replace the form (2) in this argument by any other bilinear form that obeyed the relations and for . However, this additional level of generality does not add much; any such will differ from by an antisymmetric form (so that for all , which in characteristic two implied that for all ), and such forms can always be decomposed as , where . As such, the matrices and are conjugate, with the conjugation operator being the diagonal matrix with entries at each vertex .

Remark 7(Added later) This remark combines the two previous remarks. One can view any of the matrices in Remark 6 as components of a single canonical matrix that is still of dimensions , but takes values in the Clifford algebra from Remark 5; with this “universal algebra” perspective, one no longer needs to make any arbitrary choices of form . More precisely, let denote the vector space of functions from the hypercube to the Clifford algebra; as a real vector space, this is a dimensional space, isomorphic to the direct sum of copies of , as the Clifford algebra is itself dimensional. One can then define a canonical Clifford adjacency operator on this space bywhere are the generators of . This operator can either be identified with a Clifford-valued matrix or as a real-valued matrix. In either case one still has the key algebraic relations and , ensuring that when viewed as a real matrix, half of the eigenvalues are equal to and half equal to . One can then use this matrix in place of any of the to establish Theorem 1 (noting that Schur’s test continues to work for Clifford-valued matrices because of the norm structure on ).

To relate to the real matrices , first observe that each point in the hypercube can be associated with a one-dimensional real subspace (i.e., a line) in the Clifford algebra by the formula

for any (note that this definition is well-defined even if the are out of order or contain repetitions). This can be viewed as a discrete line bundle over the hypercube. Since for any , we see that the -dimensional real linear subspace of of sections of this bundle, that is to say the space of functions such that for all , is an invariant subspace of . (Indeed, using the left-action of the Clifford algebra on , which commutes with , one can naturally identify with , with the left action of acting purely on the first factor and acting purely on the second factor.) Any trivialisation of this line bundle lets us interpret the restriction of to as a real matrix. In particular, given one of the bilinear forms from Remark 6, we can identify with by identifying any real function with the lift defined by

whenever . A somewhat tedious computation using the properties of then eventually gives the intertwining identity

and so is conjugate to .

** — 1. The Chung, Furedi, Graham, and Seymour example — **

The paper of by Chung, Furedi, Graham, and Seymour gives, for any , an example of a subset of of cardinality for which the maximum degree of restricted to is at most , thus showing that Theorem 1 cannot be improved (beyond the trivial improvement of upgrading to , because the maximum degree is obviously a natural number).

Define the “Möbius function” to be the function

for . This function is extremely balanced on coordinate spaces. Indeed, from the binomial theorem (which uses the convention ) we have

More generally, given any index set of cardinality , we have

Now let be a partition of into disjoint non-empty sets. For each , let be the subspace of consisting of those such that for all . Then for any , we have

and the right-hand side vanishes if and equals when . Applying the inclusion-exclusion principle, we conclude that

and thus also (assuming )

so that

Thus, if denotes the set of those with , together with those with , then has to have two more elements than its complement , and hence has cardinality .

Now observe that, if with and , then , and if then unless . Thus in this case the total number of for which is at most . Conversely, if with and , then , and for each there is at most one that will make lie in . Hence in this case the total number of for which is at most . Thus the maximum degree of the subgraph of induced by is at most . By choosing the to be a partition of into pieces, each of cardinality at most , we obtain the claim.

Remark 8Suppose that is a perfect square, then the lower bound here exactly matches the upper bound in Theorem 1. In particular, the minor of the matrix must have an eigenvector of eigenvalue . Such an eigenvector can be explicitly constructed as follows. Let be the vector defined by settingfor some , , , and for all other (one easily verifies that the previous types of lie in ). We claim that

for all . Expanding out the left-hand side, we wish to show that

First suppose that is of the form (3). One checks that lies in precisely when for one of the , in which case

Since , this simplifies using (3) as

giving (5) in this case. Similarly, if is of the form (4), then lies in precisely when , in which case one can argue as before to show that

and (5) again follows. Finally, if is not of either of the two forms (3), (4), one can check that is never of these forms either, and so both sides of (5) vanish.

The same analysis works for any of the other bilinear forms in Remark 6. Using the Clifford-valued operator from Remark 7, the eigenfunction is cleaner; it is defined by

when is of the form (3), and

when is of the form (4), with otherwise.

** — 2. From induced subgraph bounds to the sensitivity conjecture — **

On the hypercube , let denote the functions

The monomials in are then the characters of , so by Fourier expansion every function can be viewed as a polynomial in the (with each monomial containing at most one copy of ; higher powers of each are unnecessary since . In particular, one can meaningfully talk about the degree of a function . Observe also that the Möbius function from the preceding section is just the monomial .

Define the *sensitivity* of a Boolean function to be the largest number for which there is an such that there are at least values of with . Using an argument of Gotsman and Linial, we can now relate the sensitivity of a function to its degree:

Corollary 9 (Lower bound on sensitivity)For any boolean function , one has .

*Proof:* Write . By permuting the indices, we may assume that contains a non-trivial multiple of the monomial . By restricting to the subspace (which cannot increase the sensitivity), we may then assume without loss of generality that . The Fourier coefficient of is just the mean value

of times the Möbius function , so this mean value is non-zero. This means that one of the sets or has cardinality at least . Let denote the larger of these two sets. By Theorem 1, there is an such that for at least values of ; since , this implies that for at least values of , giving the claim.

The construction of Chung, Furedi, Graham, and Seymour from the previous section can be easily adapted to show that this lower bound is tight (other than the trivial improvement of replacing by ).

Now we need to digress on some bounds involving polynomials of one variable. We begin with an inequality of Bernstein concerning trigonometric polynomials:

Lemma 10 (Bernstein inequality)Let be a trigonometric polynomial of degree at most , that is to say a complex linear combination of for . Then

Observe that equality holds when or . Specialising to linear combinations of , we obtain the classical Bernstein inequality

for complex polynomials of degree at most .

*Proof:* If one is willing to lose a constant factor in this estimate, this bound can be easily established from modern Littlewood-Paley theory (see e.g., Exercise 52 of these lecture notes). Here we use an interlacing argument due to Boas. We first restrict to the case when has real coefficients. We may normalise . Let be a real parameter in . The trigonometric polynomial alternately takes the values and at the values . Thus the trigonometric polynomial alternates in sign at these values, and thus by the intermediate value theorem has a zero on each of the intervals . On the other hand, a trigonometric polynomial of degree at most can be expressed by de Moivre’s theorem as times a complex polynomial in of degree at most , and thus has at most zeroes. Thus we see that has exactly one zero in each . Furthermore, at this zero, the derivative of this function must be positive if is increasing on this interval, and negative if is decreasing on this interval. In summary, we have shown that if and are such that , then has the same sign as . By translating the function , we also conclude that if and are such that for some , then has the same sign as .

If , then we can find such that and is positive, and we conclude that ; thus we have the upper bound

A similar argument (with now chosen to be negative) similarly bounds . This gives the claim for real-valued trigonometric polynomials . (Indeed, this argument even gives the slightly sharper bound .)

To amplify this to complex valued polynomials, we take advantage of phase rotation invariance. If is a complex trigonometric polynomial, then by applying Bernstein’s inequality to the real part we have

But then we can multiply by any complex phase and conclude that

Taking suprema in , one obtains the claim for complex polynomials .

The analogue of Bernstein’s inequality for the unit interval is known as Markov’s inequality for polynomials:

Lemma 11 (Markov’s inequality for polynomials)Let be a polynomial of degree . Then

This bound is sharp, as is seen by inspecting the Chebyshev polynomial , defined as the unique polynomial giving the trigonometric identity

Differentiating (6) using the chain rule, we see that

the right-hand side approaches as , demonstrating that the factor here is sharp.

*Proof:* We again use an argument of Boas. We may normalise so that

The function is a trigonometric polynomial of degree at most , so by Bernstein’s inequality and the chain rule we have

for all . This already gives Markov’s inequality except in the edge regions (since ). By reflection symmetry, it then suffices to verify Markov’s inequality in the region .

From (6), the Chebyshev polynomial attains the values alternately at the different points . Thus, if , the polynomial changes sign at least times on , and thus must have all zeroes inside this interval by the intermediate value theorem; furthermore, of these zeroes will lie to the left of . By Rolle’s theorem, the derivative then has all zeroes in the interval , and at least of these will lie to the left of . In particular, the derivative can have at most one zero to once to the right of .

Since , is positive at , and hence positive as since there are no zeroes outside of . Thus the leading coefficient of is positive, which implies the same for its derivative . Thus is positive when .

From (9) one has , hence by (7) we see that is also positive at . Thus cannot become negative for , as this would create at least two zeroes to the right of . We conclude that in this region we have

From (7) we have , and the claim follows.

Remark 12The following slightly shorter argument gives the slightly weaker bound . We again use the normalisation (8). By two applications of Bernstein’s inequality, the function has first derivative bounded in magnitude by , and second derivative bounded in magnitude by . As this function also has vanishing first derivative at , we conclude the boundsand thus by the chain rule

For , one easily checks that the right-hand side is at most , giving the claim.

This implies a result of Ehlich-Zeller and of Rivlin-Cheney:

Corollary 13 (Discretised Markov inequality)Let be a polynomial of degree . If

*Proof:* We use an argument of Nisan and Szegedy. Assume for sake of contradiction that , so in particular . From the fundamental theorem of calculus and the triangle inequality one has

By a rescaled and translated version of Markov’s inequality we have

which when inserted into the preceding inequality gives after some rearranging

and then after a second application of (11) gives

Comparing with (10), we conclude that

and the claim follows after some rearranging.

Nisan and Szegedy observed that this one-dimensional degree bound can be lifted to the hypercube by a symmetrisation argument:

Corollary 14 (Multidimensional Markov inequality bound)Let be such that and for . Then

*Proof:* By averaging over all permutations of the indices (which can decrease the degree of , but not increase it), we may assume that is a symmetric function of the inputs . Using the Newton identities, we can then write

for some real polynomial of degree at most , where

is the Hamming length of . By hypothesis, , , and , hence by the mean-value theorem . Applying Corollary 13 with , we obtain the claim.

Define the *block sensitivity* of a Boolean function to be the largest number for which there is an such that there are at least disjoint subsets of with for . We have

More precisely, the sensitivity conjecture of Nisan and Szegedy asserted the bound ; Huang’s result thus gives explicit values for the exponents. It is still open whether the exponent in this theorem can be improved; it is known that one cannot improve it to below , by analysing a variant of the Chung-Furedi-Graham-Seymour example (see these notes of Regan for details). *Proof:* The lower bound for is immediate from the definitions, since the sensitivity arises by restricting the in the definition of block sensitivity to singleton sets. To prove the upper bound, it suffices from Proposition 9 to establish the bound

Let . By hypothesis, there are and disjoint subsets of such that for . We may normalise , and . If we then define the pullback boolean function by the formula

then it is easy to see that , , and for . The claim now follows from Corollary 14.

Remark 16The following slightly shorter variant of the argument lets one remove the factor . Let be as above. We again may normalise and . For any , let be iid Bernoulli random variable that equal with probability and with probability . The quantityis a trigonometric polynomial of degree at most that is bounded in magnitude by , so by two applications of Bernstein’s inequality

On the other hand, for small , the random variable is equal to zero with probability and equal to each with probability , hence

and hence . Combining these estimates we obtain and hence .

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Remark 17The sensitivity of a Boolean function can be split as , where is largest number for which there is an such that there are at least values of with . It is not difficult to use the case of Remark 3 to improve Corollary 9 slightly to . Combining this with the previous remark, we can thus improve the upper bound in Theorem 15 slightly to