whenever are compact non-empty subsets of a compact connected additive group with probability Haar measure . (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime

The connectedness of is essential, otherwise one could form counterexamples involving proper subgroups of of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as

which in turn easily follows from the identity and the inclusion , combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking to be the unit circle and to be arcs in that circle (obeying (2)). A bit more generally, if is an arbitrary connected compact abelian group and is a non-trivial character (i.e., a continuous homomorphism), then must be surjective (as has no non-trivial connected subgroups), and one can take and for some arcs in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then must be close (in measure) to an example of the above form . Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset is replaced by the partial sumset consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call a *critical pair* if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in , and invariant with respect to translation of either or . Furthermore, from the submodularity inequality (3), one can show that if and are critical pairs (with and positive), then and are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when come from arcs of a circle.) Similarly, from associativity , one can show that if and are critical pairs, then so are and .

One can combine these closure properties to obtain further ones. For instance, suppose is such that , so that all translates , intersect each other in a set of positive measure. Suppose also that is a critical pair and . Then (cheating a little bit), one can show that is also a critical pair, basically because is the union of the , , the are all critical pairs, and the all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate by the union of finitely many of the , then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair and end up with a small set such that and are also critical pairs for all (say), where is the -fold sumset of . (Intuitively, if are thought of as secretly coming from the pullback of arcs by some character , then should be the pullback of a much shorter arc by the same character.) In particular, exhibits linear growth, in that for all . One can now use standard technology from inverse sumset theory to show first that has a very large Fourier coefficient (and thus is biased with respect to some character ), and secondly that is in fact almost of the form for some arc , from which it is not difficult to conclude similar statements for and and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)

*[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman. Thanks to John Griesmer for pointing out the error.]*

Filed under: math.CO, paper Tagged: inverse theorems, Kneser's theorem, sum set estimates ]]>

- The constant function ;
- The Kronecker delta function ;
- The natural logarithm function ;
- The divisor function ;
- The von Mangoldt function , with defined to equal when is a power of a prime for some , and defined to equal zero otherwise; and
- The Möbius function , with defined to equal when is the product of distinct primes, and defined to equal zero otherwise.

Given an arithmetic function , we are often interested in statistics such as the summatory function

the logarithmically (or harmonically) weighted summatory function

or the Dirichlet series

In the latter case, one typically has to first restrict to those complex numbers whose real part is large enough in order to ensure the series on the right converges; but in many important cases, one can then extend the Dirichlet series to almost all of the complex plane by analytic continuation. One is also interested in correlations involving additive shifts, such as , but these are significantly more difficult to study and cannot be easily estimated by the methods of classical multiplicative number theory.

A key operation on arithmetic functions is that of Dirichlet convolution, which when given two arithmetic functions , forms a new arithmetic function , defined by the formula

Thus for instance , , , and for any arithmetic function . Dirichlet convolution and Dirichlet series are related by the fundamental formula

at least when the real part of is large enough that all sums involved become absolutely convergent (but in practice one can use analytic continuation to extend this identity to most of the complex plane). There is also the identity

at least when the real part of is large enough to justify interchange of differentiation and summation. As a consequence, many Dirichlet series can be expressed in terms of the Riemann zeta function , thus for instance

Much of the difficulty of multiplicative number theory can be traced back to the discrete nature of the natural numbers , which form a rather complicated abelian semigroup with respect to multiplication (in particular the set of generators is the set of prime numbers). One can obtain a simpler analogue of the subject by working instead with the half-infinite interval , which is a much simpler abelian semigroup under multiplication (being a one-dimensional Lie semigroup). (I will think of this as a sort of “completion” of at the infinite place , hence the terminology.) Accordingly, let us define a *continuous arithmetic function* to be a locally integrable function . The analogue of the summatory function (1) is then an integral

and similarly the analogue of (2) is

The analogue of the Dirichlet series is the Mellin-type transform

which will be well-defined at least if the real part of is large enough and if the continuous arithmetic function does not grow too quickly, and hopefully will also be defined elsewhere in the complex plane by analytic continuation.

For instance, the continuous analogue of the discrete constant function would be the constant function , which maps any to , and which we will denote by in order to keep it distinct from . The two functions and have approximately similar statistics; for instance one has

and

where is the harmonic number, and we are deliberately vague as to what the symbol means. Continuing this analogy, we would expect

which reflects the fact that has a simple pole at with residue , and no other poles. Note that the identity is initially only valid in the region , but clearly the right-hand side can be continued analytically to the entire complex plane except for the pole at , and so one can define in this region also.

In a similar vein, the logarithm function is approximately similar to the logarithm function , giving for instance the crude form

of Stirling’s formula, or the Dirichlet series approximation

The continuous analogue of Dirichlet convolution is multiplicative convolution using the multiplicative Haar measure : given two continuous arithmetic functions , one can define their convolution by the formula

Thus for instance . A short computation using Fubini’s theorem shows the analogue

of (3) whenever the real part of is large enough that Fubini’s theorem can be justified; similarly, differentiation under the integral sign shows that

again assuming that the real part of is large enough that differentiation under the integral sign (or some other tool like this, such as the Cauchy integral formula for derivatives) can be justified.

Direct calculation shows that for any complex number , one has

(at least for the real part of large enough), and hence by several applications of (5)

for any natural number . This can lead to the following heuristic: if a Dirichlet series behaves like a linear combination of poles , in that

for some set of poles and some coefficients and natural numbers (where we again are vague as to what means, and how to interpret the sum if the set of poles is infinite), then one should expect the arithmetic function to behave like the continuous arithmetic function

In particular, if we only have simple poles,

then we expect to have behave like continuous arithmetic function

Integrating this from to , this heuristically suggests an approximation

for the summatory function, and similarly

with the convention that is when , and similarly is when . One can make these sorts of approximations more rigorous by means of Perron’s formula (or one of its variants) combined with the residue theorem, provided that one has good enough control on the relevant Dirichlet series, but we will not pursue these rigorous calculations here. (But see for instance this previous blog post for some examples.)

For instance, using the more refined approximation

to the zeta function near , we have

we would expect that

and thus for instance

which matches what one actually gets from the Dirichlet hyperbola method (see e.g. equation (44) of this previous post).

Or, noting that has a simple pole at and assuming simple zeroes elsewhere, the log derivative will have simple poles of residue at and at all the zeroes, leading to the heuristic

suggesting that should behave like the continuous arithmetic function

leading for instance to the summatory approximation

which is a heuristic form of the Riemann-von Mangoldt explicit formula (see Exercise 45 of these notes for a rigorous version of this formula).

Exercise 1Go through some of the other explicit formulae listed at this Wikipedia page and give heuristic justifications for them (up to some lower order terms) by similar calculations to those given above.

Given the “adelic” perspective on number theory, I wonder if there are also -adic analogues of arithmetic functions to which a similar set of heuristics can be applied, perhaps to study sums such as . A key problem here is that there does not seem to be any good interpretation of the expression when is complex and is a -adic number, so it is not clear that one can analyse a Dirichlet series -adically. For similar reasons, we don’t have a canonical way to define for a Dirichlet character (unless its conductor happens to be a power of ), so there doesn’t seem to be much to say in the -aspect either.

Filed under: expository, math.NT Tagged: Dirichlet series, prime number theorem ]]>

While we already have several confirmed participants, we are still accepting applications for this program until Dec 4; details of the application process may be found at this page.

Filed under: advertising Tagged: ipam, linear algebra ]]>

More information and an application form for the scholarship can be found on the web at:

http://www.math.ucla.edu/ugrad/mums

To be considered for Fall 2018, candidates must apply for the scholarship and also for admission to UCLA on or before November 30, 2017.

Filed under: advertising Tagged: scholarship, UCLA, undergraduate study ]]>

for all and all distinct natural numbers , where we use the averaging notation

For , this conjecture is equivalent to the prime number theorem (as discussed in this previous blog post), but the conjecture remains open for any .

In recent years, it has been realised that one can make more progress on this conjecture if one works instead with the logarithmically averaged version

of the conjecture, where we use the logarithmic averaging notation

Using the summation by parts (or telescoping series) identity

it is not difficult to show that the Chowla conjecture (1) for a given implies the logarithmically averaged conjecture (2). However, the converse implication is not at all clear. For instance, for , we have already mentioned that the Chowla conjecture

is equivalent to the prime number theorem; but the logarithmically averaged analogue

is significantly easier to show (a proof with the Liouville function replaced by the closely related Möbius function is given in this previous blog post). And indeed, significantly more is now known for the logarithmically averaged Chowla conjecture; in this paper of mine I had proven (2) for , and in this recent paper with Joni Teravainen, we proved the conjecture for all odd (with a different proof also given here).

In view of this emerging consensus that the logarithmically averaged Chowla conjecture was easier than the ordinary Chowla conjecture, it was thus somewhat of a surprise for me to read a recent paper of Gomilko, Kwietniak, and Lemanczyk who (among other things) established the following statement:

Theorem 1Assume that the logarithmically averaged Chowla conjecture (2) is true for all . Then there exists a sequence going to infinity such that the Chowla conjecture (1) is true for all along that sequence, that is to sayfor all and all distinct .

This implication does not use any special properties of the Liouville function (other than that they are bounded), and in fact proceeds by ergodic theoretic methods, focusing in particular on the ergodic decomposition of invariant measures of a shift into ergodic measures. Ergodic methods have proven remarkably fruitful in understanding these sorts of number theoretic and combinatorial problems, as could already be seen by the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recently by the work of Frantzikinakis and Host on Sarnak’s conjecture. (My first paper with Teravainen also uses ergodic theory tools.) Indeed, many other results in the subject were first discovered using ergodic theory methods.

On the other hand, many results in this subject that were first proven ergodic theoretically have since been reproven by more combinatorial means; my second paper with Teravainen is an instance of this. As it turns out, one can also prove Theorem 1 by a standard combinatorial (or probabilistic) technique known as the second moment method. In fact, one can prove slightly more:

Theorem 2Let be a natural number. Assume that the logarithmically averaged Chowla conjecture (2) is true for . Then there exists a set of natural numbers of logarithmic density (that is, ) such thatfor any distinct .

It is not difficult to deduce Theorem 1 from Theorem 2 using a diagonalisation argument. Unfortunately, the known cases of the logarithmically averaged Chowla conjecture ( and odd ) are currently insufficient to use Theorem 2 for any purpose other than to reprove what is already known to be true from the prime number theorem. (Indeed, the even cases of Chowla, in either logarithmically averaged or non-logarithmically averaged forms, seem to be far more powerful than the odd cases; see Remark 1.7 of this paper of myself and Teravainen for a related observation in this direction.)

We now sketch the proof of Theorem 2. For any distinct , we take a large number and consider the limiting the second moment

We can expand this as

If all the are distinct, the hypothesis (2) tells us that the inner averages goes to zero as . The remaining averages are , and there are of these averages. We conclude that

By Markov’s inequality (and (3)), we conclude that for any fixed , there exists a set of upper logarithmic density at least , thus

such that

By deleting at most finitely many elements, we may assume that consists only of elements of size at least (say).

For any , if we let be the union of for , then has logarithmic density . By a diagonalisation argument (using the fact that the set of tuples is countable), we can then find a set of natural numbers of logarithmic density , such that for every , every sufficiently large element of lies in . Thus for every sufficiently large in , one has

for some with . By Cauchy-Schwarz, this implies that

interchanging the sums and using and , this implies that

We conclude on taking to infinity that

as required.

Filed under: expository, math.CO, math.DS, math.NT, math.PR Tagged: Chowla conjecture, second moment method ]]>

for some complex zeroes (possibly with repetition).

Now suppose we evolve with respect to time by heat flow, creating a function of two variables with given initial data for which

On the space of polynomials of degree at most , the operator is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series

For instance, if one starts with a quadratic , then the polynomial evolves by the formula

As the polynomial evolves in time, the zeroes evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution?

For instance, in the quadratic case, the quadratic formula tells us that the zeroes are

and

after arbitrarily choosing a branch of the square root. If are real and the discriminant is initially positive, we see that we start with two real zeroes centred around , which then approach each other until time , at which point the roots collide and then move off from each other in an imaginary direction.

In the general case, we can obtain the equations of motion by implicitly differentiating the defining equation

in time using (2) to obtain

To simplify notation we drop the explicit dependence on time, thus

From (1) and the product rule, we see that

and

(where all indices are understood to range over ) leading to the equations of motion

at least when one avoids those times in which there is a repeated zero. In the case when the zeroes are real, each term represents a (first-order) attraction in the dynamics between and , but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of Dyson brownian motion (except with the brownian motion part removed, and time reversed). I learned of the connection between the ODE (3) and the heat equation from this paper of Csordas, Smith, and Varga, but perhaps it has been mentioned in earlier literature as well.

One interesting consequence of these equations is that if the zeroes are real at some time, then they will stay real as long as the zeroes do not collide. Let us now restrict attention to the case of real simple zeroes, in which case we will rename the zeroes as instead of , and order them as . The evolution

can now be thought of as reverse gradient flow for the “entropy”

(which is also essentially the logarithm of the discriminant of the polynomial) since we have

In particular, we have the monotonicity formula

where is the “energy”

where in the last line we use the antisymmetrisation identity

Among other things, this shows that as one goes backwards in time, the entropy decreases, and so no collisions can occur to the past, only in the future, which is of course consistent with the attractive nature of the dynamics. As is a convex function of the positions , one expects to also evolve in a convex manner in time, that is to say the energy should be increasing. This is indeed the case:

Exercise 1Show that

Symmetric polynomials of the zeroes are polynomial functions of the coefficients and should thus evolve in a polynomial fashion. One can compute this explicitly in simple cases. For instance, the center of mass is an invariant:

The variance decreases linearly:

Exercise 2Establish the virial identity

As the variance (which is proportional to ) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time .

Exercise 3Show that theStieltjes transformsolves the viscous Burgers equation

either by using the original heat equation (2) and the identity , or else by using the equations of motion (3). This relation between the Burgers equation and the heat equation is known as the Cole-Hopf transformation.

The paper of Csordas, Smith, and Varga mentioned previously gives some other bounds on the lifespan of the dynamics; roughly speaking, they show that if there is one pair of zeroes that are much closer to each other than to the other zeroes then they must collide in a short amount of time (unless there is a collision occuring even earlier at some other location). Their argument extends also to situations where there are an infinite number of zeroes, which they apply to get new results on Newman’s conjecture in analytic number theory. I would be curious to know of further places in the literature where this dynamics has been studied.

Filed under: expository, math.AP, math.CV, math.DS Tagged: entropy, heat flow, polynomials, zeroes ]]>

for all odd and all integers (that is to say, all the odd order cases of the logarithmically averaged Chowla conjecture). Our previous argument relies heavily on some deep ergodic theory results of Bergelson-Host-Kra, Leibman, and Le (and was applicable to more general multiplicative functions than the Liouville function ); here we give a shorter proof that avoids ergodic theory (but instead requires the Gowers uniformity of the (W-tricked) von Mangoldt function, established in several papers of Ben Green, Tamar Ziegler, and myself). The proof follows the lines sketched in the previous blog post. In principle, due to the avoidance of ergodic theory, the arguments here have a greater chance to be made quantitative; however, at present the known bounds on the Gowers uniformity of the von Mangoldt function are qualitative, except at the level, which is unfortunate since the first non-trivial odd case requires quantitative control on the level. (But it may be possible to make the Gowers uniformity bounds for quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)

Filed under: math.NT, paper Tagged: Chowla conjecture, Joni Teravainen ]]>

of skew-Schur polynomials is Schur-positive for any partitions (using the convention that the skew-Schur polynomial vanishes if is not contained in , and where and denotes the pointwise min and max of and respectively). It is fairly easy to derive the monotonicity of from this, by using the expansion

of Schur polynomials into skew-Schur polynomials (as was done in this previous post).

The second proof of monotonicity avoids representation theory by a more elementary argument establishing the weaker claim that the above expression (1) is non-negative on the positive orthant. In fact we prove a more general determinantal log-supermodularity claim which may be of independent interest:

Theorem 1Let be any totally positive matrix (thus, every minor has a non-negative determinant). Then for any -tuples of increasing elements of , one haswhere denotes the minor formed from the rows in and columns in .

For instance, if is the matrix

for some real numbers , one has

(corresponding to the case , ), or

(corresponding to the case , , , , ). It turns out that this claim can be proven relatively easy by an induction argument, relying on the Dodgson and Karlin identities from this previous post; the difficulties are largely notational in nature. Combining this result with the Jacobi-Trudi identity for skew-Schur polynomials (discussed in this previous post) gives the non-negativity of (1); it can also be used to directly establish the monotonicity of ratios by applying the theorem to a generalised Vandermonde matrix.

(Log-supermodularity also arises as the natural hypothesis for the FKG inequality, though I do not know of any interesting application of the FKG inequality in this current setting.)

Filed under: math.CA, math.RA, update Tagged: Apoorva Khare, Dodgson condensation, Schur polynomials, supermodularity, total positivity ]]>

where is an matrix, is an matrix, is an matrix, and is a matrix for some . If is invertible, we can use the technique of Schur complementation to express the inverse of (if it exists) in terms of the inverse of , and the other components of course. Indeed, to solve the equation

where are column vectors and are column vectors, we can expand this out as a system

Using the invertibility of , we can write the first equation as

and substituting this into the second equation yields

and thus (assuming that is invertible)

and then inserting this back into (1) gives

Comparing this with

we have managed to express the inverse of as

One can consider the inverse problem: given the inverse of , does one have a nice formula for the inverse of the minor ? Trying to recover this directly from (2) looks somewhat messy. However, one can proceed as follows. Let denote the matrix

(with the identity matrix), and let be its transpose:

Then for any scalar (which we identify with times the identity matrix), one has

and hence by (2)

noting that the inverses here will exist for large enough. Taking limits as , we conclude that

On the other hand, by the Woodbury matrix identity (discussed in this previous blog post), we have

and hence on taking limits and comparing with the preceding identity, one has

This achieves the aim of expressing the inverse of the minor in terms of the inverse of the full matrix. Taking traces and rearranging, we conclude in particular that

In the case, this can be simplified to

where is the basis column vector.

We can apply this identity to understand how the spectrum of an random matrix relates to that of its top left minor . Subtracting any complex multiple of the identity from (and hence from ), we can relate the Stieltjes transform of with the Stieltjes transform of :

At this point we begin to proceed informally. Assume for sake of argument that the random matrix is Hermitian, with distribution that is invariant under conjugation by the unitary group ; for instance, could be drawn from the Gaussian Unitary Ensemble (GUE), or alternatively could be of the form for some real diagonal matrix and a unitary matrix drawn randomly from using Haar measure. To fix normalisations we will assume that the eigenvalues of are typically of size . Then is also Hermitian and -invariant. Furthermore, the law of will be the same as the law of , where is now drawn uniformly from the unit sphere (independently of ). Diagonalising into eigenvalues and eigenvectors , we have

One can think of as a random (complex) Gaussian vector, divided by the magnitude of that vector (which, by the Chernoff inequality, will concentrate to ). Thus the coefficients with respect to the orthonormal basis can be thought of as independent (complex) Gaussian vectors, divided by that magnitude. Using this and the Chernoff inequality again, we see (for distance away from the real axis at least) that one has the concentration of measure

and thus

(that is to say, the diagonal entries of are roughly constant). Similarly we have

Inserting this into (5) and discarding terms of size , we thus conclude the approximate relationship

This can be viewed as a difference equation for the Stieltjes transform of top left minors of . Iterating this equation, and formally replacing the difference equation by a differential equation in the large limit, we see that when is large and for some , one expects the top left minor of to have Stieltjes transform

where solves the Burgers-type equation

Example 1If is a constant multiple of the identity, then . One checks that is a steady state solution to (7), which is unsurprising given that all minors of are also times the identity.

Example 2If is GUE normalised so that each entry has variance , then by the semi-circular law (see previous notes) one has (using an appropriate branch of the square root). One can then verify the self-similar solutionto (7), which is consistent with the fact that a top minor of also has the law of GUE, with each entry having variance when .

One can justify the approximation (6) given a sufficiently good well-posedness theory for the equation (7). We will not do so here, but will note that (as with the classical inviscid Burgers equation) the equation can be solved exactly (formally, at least) by the method of characteristics. For any initial position , we consider the characteristic flow formed by solving the ODE

with initial data , ignoring for this discussion the problems of existence and uniqueness. Then from the chain rule, the equation (7) implies that

and thus . Inserting this back into (8) we see that

and thus (7) may be solved implicitly via the equation

Remark 3In practice, the equation (9) may stop working when crosses the real axis, as (7) does not necessarily hold in this region. It is a cute exercise (ultimately coming from the Cauchy-Schwarz inequality) to show that this crossing always happens, for instance if has positive imaginary part then necessarily has negative or zero imaginary part.

Example 4Suppose we have as in Example 1. Then (9) becomesfor any , which after making the change of variables becomes

as in Example 1.

Example 5Suppose we haveas in Example 2. Then (9) becomes

If we write

one can calculate that

and hence

One can recover the spectral measure from the Stieltjes transform as the weak limit of as ; we write this informally as

In this informal notation, we have for instance that

which can be interpreted as the fact that the Cauchy distributions converge weakly to the Dirac mass at as . Similarly, the spectral measure associated to (10) is the semicircular measure .

If we let be the spectral measure associated to , then the curve from to the space of measures is the high-dimensional limit of a Gelfand-Tsetlin pattern (discussed in this previous post), if the pattern is randomly generated amongst all matrices with spectrum asymptotic to as . For instance, if , then the curve is , corresponding to a pattern that is entirely filled with ‘s. If instead is a semicircular distribution, then the pattern is

thus at height from the top, the pattern is semicircular on the interval . The interlacing property of Gelfand-Tsetlin patterns translates to the claim that (resp. ) is non-decreasing (resp. non-increasing) in for any fixed . In principle one should be able to establish these monotonicity claims directly from the PDE (7) or from the implicit solution (9), but it was not clear to me how to do so.

An interesting example of such a limiting Gelfand-Tsetlin pattern occurs when , which corresponds to being , where is an orthogonal projection to a random -dimensional subspace of . Here we have

and so (9) in this case becomes

A tedious calculation then gives the solution

For , there are simple poles at , and the associated measure is

This reflects the interlacing property, which forces of the eigenvalues of the minor to be equal to (resp. ). For , the poles disappear and one just has

For , one has an inverse semicircle distribution

There is presumably a direct geometric explanation of this fact (basically describing the singular values of the product of two random orthogonal projections to half-dimensional subspaces of ), but I do not know of one off-hand.

The evolution of can also be understood using the *-transform* and *-transform* from free probability. Formally, letlet be the inverse of , thus

for all , and then define the -transform

The equation (9) may be rewritten as

and hence

See these previous notes for a discussion of free probability topics such as the -transform.

Example 6If then the transform is .

Example 7If is given by (10), then the transform is

Example 8If is given by (11), then the transform is

This simple relationship (12) is essentially due to Nica and Speicher (thanks to Dima Shylakhtenko for this reference). It has the remarkable consequence that when is the reciprocal of a natural number , then is the free arithmetic mean of copies of , that is to say is the free convolution of copies of , pushed forward by the map . In terms of random matrices, this is asserting that the top minor of a random matrix has spectral measure approximately equal to that of an arithmetic mean of independent copies of , so that the process of taking top left minors is in some sense a continuous analogue of the process of taking freely independent arithmetic means. There ought to be a geometric proof of this assertion, but I do not know of one. In the limit (or ), the -transform becomes linear and the spectral measure becomes semicircular, which is of course consistent with the free central limit theorem.

In a similar vein, if one defines the function

and inverts it to obtain a function with

for all , then the *-transform* is defined by

Writing

for any , , we have

and so (9) becomes

which simplifies to

replacing by we obtain

and thus

and hence

One can compute to be the -transform of the measure ; from the link between -transforms and free products (see e.g. these notes of Guionnet), we conclude that is the free product of and . This is consistent with the random matrix theory interpretation, since is also the spectral measure of , where is the orthogonal projection to the span of the first basis elements, so in particular has spectral measure . If is unitarily invariant then (by a fundamental result of Voiculescu) it is asymptotically freely independent of , so the spectral measure of is asymptotically the free product of that of and of .

Filed under: expository, math.PR, math.RA, math.SP Tagged: free probability, Gelfand-Tsetlin patterns, Schur complement ]]>

Back in 2005, I rewrote Szemerédi’s original proof in order to understand it better, however my rewrite ended up being about the same length as the original argument and was probably only usable to myself. In 2012, after Szemerédi was awarded the Abel prize, I revisited this argument with the intention to try to write up a more readable version of the proof, but ended up just presenting some ingredients of the argument in a blog post, rather than try to rewrite the whole thing. In that post, I suspected that the cleanest way to write up the argument would be through the language of nonstandard analysis (perhaps in an iterated hyperextension that could handle various hierarchies of infinitesimals), but was unable to actually achieve any substantial simplifications by passing to the nonstandard world.

A few weeks ago, I participated in a week-long workshop at the American Institute of Mathematics on “Nonstandard methods in combinatorial number theory”, and spent some time in a working group with Shabnam Akhtari, Irfam Alam, Renling Jin, Steven Leth, Karl Mahlburg, Paul Potgieter, and Henry Towsner to try to obtain a manageable nonstandard version of Szemerédi’s original proof. We didn’t end up being able to do so – in fact there are now signs that perhaps nonstandard analysis is not the optimal framework in which to place this argument – but we did at least clarify the existing standard argument, to the point that I was able to go back to my original rewrite of the proof and present it in a more civilised form, which I am now uploading here as an unpublished preprint. There are now a number of simplifications to the proof. Firstly, one no longer needs the full strength of the regularity lemma; only the simpler “weak” regularity lemma of Frieze and Kannan is required. Secondly, the proof has been “factored” into a number of stand-alone propositions of independent interest, in particular involving just (families of) one-dimensional arithmetic progressions rather than the complicated-looking multidimensional arithmetic progressions that occur so frequently in the original argument of Szemerédi. Finally, the delicate manipulations of densities and epsilons via double counting arguments in Szemerédi’s original paper have been abstracted into a certain key property of families of arithmetic progressions that I call the “double counting property”.

The factoring mentioned above is particularly simple in the case of proving Roth’s theorem, which is now presented separately in the above writeup. Roth’s theorem seeks to locate a length three progression in which all three elements lie in a single set. This will be deduced from an easier variant of the theorem in which one locates (a family of) length three progressions in which just the first two elements of the progression lie in a good set (and some other properties of the family are also required). This is in turn derived from an even easier variant in which now just the first element of the progression is required to be in the good set.

More specifically, Roth’s theorem is now deduced from

Theorem 1.5. Let be a natural number, and let be a set of integers of upper density at least . Then, whenever is partitioned into finitely many colour classes, there exists a colour class and a family of 3-term arithmetic progressions with the following properties:

- For each , and lie in .
- For each , lie in .
- The for are in arithmetic progression.

The situation in this theorem is depicted by the following diagram, in which elements of are in blue and elements of are in grey:

Theorem 1.5 is deduced in turn from the following easier variant:

Theorem 1.6. Let be a natural number, and let be a set of integers of upper density at least . Then, whenever is partitioned into finitely many colour classes, there exists a colour class and a family of 3-term arithmetic progressions with the following properties:

- For each , lie in .
- For each , and lie in .
- The for are in arithmetic progression.

The situation here is described by the figure below.

Theorem 1.6 is easy to prove. To derive Theorem 1.5 from Theorem 1.6, or to derive Roth’s theorem from Theorem 1.5, one uses double counting arguments, van der Waerden’s theorem, and the weak regularity lemma, largely as described in this previous blog post; see the writeup for the full details. (I would be interested in seeing a shorter proof of Theorem 1.5 though that did not go through these arguments, and did not use the more powerful theorems of Roth or Szemerédi.)

Filed under: expository, math.CO Tagged: regularity lemma, Roth's theorem, Szemeredi's theorem ]]>