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Just a brief post to record some notable papers in my fields of interest that appeared on the arXiv recently.
- “A sharp square function estimate for the cone in
“, by Larry Guth, Hong Wang, and Ruixiang Zhang. This paper establishes an optimal (up to epsilon losses) square function estimate for the three-dimensional light cone that was essentially conjectured by Mockenhaupt, Seeger, and Sogge, which has a number of other consequences including Sogge’s local smoothing conjecture for the wave equation in two spatial dimensions, which in turn implies the (already known) Bochner-Riesz, restriction, and Kakeya conjectures in two dimensions. Interestingly, modern techniques such as polynomial partitioning and decoupling estimates are not used in this argument; instead, the authors mostly rely on an induction on scales argument and Kakeya type estimates. Many previous authors (including myself) were able to get weaker estimates of this type by an induction on scales method, but there were always significant inefficiencies in doing so; in particular knowing the sharp square function estimate at smaller scales did not imply the sharp square function estimate at the given larger scale. The authors here get around this issue by finding an even stronger estimate that implies the square function estimate, but behaves significantly better with respect to induction on scales.
- “On the Chowla and twin primes conjectures over
“, by Will Sawin and Mark Shusterman. This paper resolves a number of well known open conjectures in analytic number theory, such as the Chowla conjecture and the twin prime conjecture (in the strong form conjectured by Hardy and Littlewood), in the case of function fields where the field is a prime power
which is fixed (in contrast to a number of existing results in the “large
” limit) but has a large exponent
. The techniques here are orthogonal to those used in recent progress towards the Chowla conjecture over the integers (e.g., in this previous paper of mine); the starting point is an algebraic observation that in certain function fields, the Mobius function behaves like a quadratic Dirichlet character along certain arithmetic progressions. In principle, this reduces problems such as Chowla’s conjecture to problems about estimating sums of Dirichlet characters, for which more is known; but the task is still far from trivial.
- “Bounds for sets with no polynomial progressions“, by Sarah Peluse. This paper can be viewed as part of a larger project to obtain quantitative density Ramsey theorems of Szemeredi type. For instance, Gowers famously established a relatively good quantitative bound for Szemeredi’s theorem that all dense subsets of integers contain arbitrarily long arithmetic progressions
. The corresponding question for polynomial progressions
is considered more difficult for a number of reasons. One of them is that dilation invariance is lost; a dilation of an arithmetic progression is again an arithmetic progression, but a dilation of a polynomial progression will in general not be a polynomial progression with the same polynomials
. Another issue is that the ranges of the two parameters
are now at different scales. Peluse gets around these difficulties in the case when all the polynomials
have distinct degrees, which is in some sense the opposite case to that considered by Gowers (in particular, she avoids the need to obtain quantitative inverse theorems for high order Gowers norms; which was recently obtained in this integer setting by Manners but with bounds that are probably not strong enough to for the bounds in Peluse’s results, due to a degree lowering argument that is available in this case). To resolve the first difficulty one has to make all the estimates rather uniform in the coefficients of the polynomials
, so that one can still run a density increment argument efficiently. To resolve the second difficulty one needs to find a quantitative concatenation theorem for Gowers uniformity norms. Many of these ideas were developed in previous papers of Peluse and Peluse-Prendiville in simpler settings.
- “On blow up for the energy super critical defocusing non linear Schrödinger equations“, by Frank Merle, Pierre Raphael, Igor Rodnianski, and Jeremie Szeftel. This paper (when combined with two companion papers) resolves a long-standing problem as to whether finite time blowup occurs for the defocusing supercritical nonlinear Schrödinger equation (at least in certain dimensions and nonlinearities). I had a previous paper establishing a result like this if one “cheated” by replacing the nonlinear Schrodinger equation by a system of such equations, but remarkably they are able to tackle the original equation itself without any such cheating. Given the very analogous situation with Navier-Stokes, where again one can create finite time blowup by “cheating” and modifying the equation, it does raise hope that finite time blowup for the incompressible Navier-Stokes and Euler equations can be established… In fact the connection may not just be at the level of analogy; a surprising key ingredient in the proofs here is the observation that a certain blowup ansatz for the nonlinear Schrodinger equation is governed by solutions to the (compressible) Euler equation, and finite time blowup examples for the latter can be used to construct finite time blowup examples for the former.
Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions. Roth’s theorem is the special case when one considers arithmetic progressions of length three. Both theorems have many important proofs using tools from additive combinatorics, (higher order) Fourier analysis, (hyper) graph regularity theory, and ergodic theory. However, the original proof by Endre Szemerédi, while extremely intricate, was purely combinatorial (and in particular “elementary”) and almost entirely self-contained, except for an invocation of the van der Waerden theorem. It is also notable for introducing a prototype of what is now known as the Szemerédi regularity lemma.
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.)
Ben Green and I have (finally!) uploaded to the arXiv our paper “New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for “, submitted to Mathematika. This is the sequel to two previous papers (and an erratum to the former paper), concerning quantitative versions of Szemerédi’s theorem in the case of length four progressions. This sequel has been delayed for over a decade for a number of reasons, but we have finally managed to write the arguments up to our satisfaction and submit it (to a special issue of Mathematika honouring the work of Klaus Roth).
For any natural number , define
to be the largest cardinality of a subset
of
which does not contain any non-trivial arithmetic progressions
of length four (where “non-trivial” means that
is non-zero). Trivially we have
. In 1969, Szemerédi showed that
. However, the decay rate that could be theoretically extracted from this argument (and from several subsequent proofs of this bound, including one by Roth) were quite poor. The first significant quantitative bound on this quantity was by Gowers, who showed that
for some absolute constant
. In the second paper in the above-mentioned series, we managed to improve this bound to
. In this paper, we improve the bound further to
, which seems to be the limit of the methods. (We remark that if we could take
to be larger than one, this would imply the length four case of a well known conjecture of Erdös that any set of natural numbers whose sum of reciprocals diverges would contain arbitrarily long arithmetic progressions. Thanks to the work of Sanders and of Bloom, the corresponding case of the conjecture for length three conjectures is nearly settled, as it is known that for the analogous bound on
one can take any
less than one.)
Most of the previous work on bounding relied in some form or another on the density increment argument introduced by Roth back in 1953; roughly speaking, the idea is to show that if a dense subset
of
fails to contain arithmetic progressions of length four, one seeks to then locate a long subprogression of
in which
has increased density. This was the basic method for instance underlying our previous bound
, as well as a finite field analogue of the bound
; however we encountered significant technical difficulties for several years in extending this argument to obtain the result of the current paper. Our method is instead based on “energy increment arguments”, and more specifically on establishing quantitative version of a Khintchine-type recurrence theorem, similar to the qualitative recurrence theorems established (in the ergodic theory context) by Bergelson-Host-Kra, and (in the current combinatorial context) by Ben Green and myself.
One way to phrase the latter recurrence theorem is as follows. Suppose that has density
. Then one would expect a “randomly” selected arithmetic progression
in
(using the convention that random variables will be in boldface) to be contained in
with probability about
. This is not true in general, however it was shown by Ben and myself that for any
, there was a set of shifts
of cardinality
, such that for any such
one had
if was chosen uniformly at random from
. This easily implies that
, but does not give a particularly good bound on the decay rate, because the implied constant in the cardinality lower bound
is quite poor (in fact of tower-exponential type, due to the use of regularity lemmas!), and so one has to take
to be extremely large compared to
to avoid the possibility that the set of shifts in the above theorem consists only of the trivial shift
.
We do not know how to improve the lower bound on the set of shifts to the point where it can give bounds that are competitive with those in this paper. However, we can obtain better quantitative results if we permit ourselves to couple together the two parameters and
of the length four progression. Namely, with
,
,
as above, we are able to show that there exist random variables
, not necessarily independent, such that
and such that we have the non-degeneracy bound
This then easily implies the main theorem.
The energy increment method is then deployed to locate a good pair of random variables that will obey the above bounds. One can get some intuition on how to proceed here by considering some model cases. Firstly one can consider a “globally quadratically structured” case in which the indicator function
“behaves like” a globally quadratic function such as
, for some irrational
and some smooth periodic function
of mean
. If one then takes
to be uniformly distributed in
and
respectively for some small
, with no coupling between the two variables, then the left-hand side of (1) is approximately of the form
where the integral is with respect to the probability Haar measure, and the constraint ultimately arises from the algebraic constraint
However, an application of the Cauchy-Schwarz inequality and Fubini’s theorem shows that the integral in (2) is at least , which (morally at least) gives (1) in this case.
Due to the nature of the energy increment argument, it also becomes necessary to consider “locally quadratically structured” cases, in which is partitioned into some number of structured pieces
(think of these as arithmetic progressions, or as “Bohr sets), and on each piece
,
behaves like a locally quadratic function such as
, where
now varies with
, and the mean of
will be approximately
on the average after averaging in
(weighted by the size of the pieces
). Now one should select
and
in the following coupled manner: first one chooses
uniformly from
, then one defines
to be the label
such that
, and then selects
uniformly from a set
which is related to
in much the same way that
is related to
. If one does this correctly, the analogue of (2) becomes
and one can again use Cauchy-Schwarz and Fubini’s theorem to conclude.
The general case proceeds, very roughly, by an iterative argument. At each stage of the iteration, one has some sort of quadratic model of which involves a decomposition of
into structured pieces
, and a quadratic approximation to
on each piece. If this approximation is accurate enough (or more precisely, if a certain (averaged) local Gowers uniformity norm
of the error is small enough) to model the count in (1) (for random variables
determined by the above partition of
into pieces
), and if the frequencies (such as
) involved in the quadratic approximation are “high rank” or “linearly independent over the rationals” in a suitably quantitative sense, then some version of the above arguments can be made to work. If there are some unwanted linear dependencies in the frequencies, we can do some linear algebra to eliminate one of the frequencies (using some geometry of numbers to keep the quantitative bounds under control) and continue the iteration. If instead the approximation is too inaccurate, then the error will be large in a certain averaged local Gowers uniformity norm
. A significant fraction of the paper is then devoted to establishing a quantitative inverse theorem for that norm that concludes (with good bounds) that the error must then locally correlate with locally quadratic phases, which can be used to refine the quadratic approximation to
in a manner that significantly increases its “energy” (basically an
norm). Such energy increments cannot continue indefinitely, and when they terminate we obtain the desired claim.
There are existing inverse theorems for type norms in the literature, going back to the pioneering work of Gowers mentioned previously, and relying on arithmetic combinatorics tools such as Freiman’s theorem and the Balog-Szemerédi-Gowers lemma, which are good for analysing the “
-structured homomorphisms” that arise in Gowers’ argument. However, when we applied these methods to the local Gowers norms we obtained inferior quantitative results that were not strong enough for our application. Instead, we use arguments from a different paper of Gowers in which he tackled Szemerédi’s theorem for arbitrary length progressions. This method produces “
-structured homomorphisms” associated to any function with large Gowers uniformity norm; however the catch is that such homomorphisms are initially supported only on a sparse unstructured set, rather than a structured set such as a Bohr set. To proceed further, one first has to locate inside the sparse unstructured set a sparse pseudorandom subset of a Bohr set, and then use “error-correction” type methods (such as “majority-vote” based algorithms) to locally upgrade this
-structured homomorphism on pseudorandom subsets of Bohr sets to a
-structured homomorphism on the entirety of a Bohr set. It is then possible to use some “approximate cohomology” tools to “integrate” these homomorphisms (and discern a key “local symmetry” property of these homomorphisms) to locate the desired local quadratic structure (in much the same fashion that a
-form on
that varies linearly with the coordinates can be integrated to be the derivative of a quadratic function if we know that the
-form is closed). These portions of the paper are unfortunately rather technical, but broadly follow the methods already used in previous literature.
Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:
Theorem 1 (Szemerédi’s theorem) Let
be a positive integer, and let
be a function with
for some
, where we use the averaging notation
,
, etc.. Then for
we have
for some
depending only on
.
The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases as they are trivial and somewhat degenerate.
There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.
Tamar Ziegler and I have just uploaded to the arXiv our joint paper “A multi-dimensional Szemerédi theorem for the primes via a correspondence principle“. This paper is related to an earlier result of Ben Green and mine in which we established that the primes contain arbitrarily long arithmetic progressions. Actually, in that paper we proved a more general result:
Theorem 1 (Szemerédi’s theorem in the primes) Let
be a subset of the primes
of positive relative density, thus
. Then
contains arbitrarily long arithmetic progressions.
This result was based in part on an earlier paper of Green that handled the case of progressions of length three. With the primes replaced by the integers, this is of course the famous theorem of Szemerédi.
Szemerédi’s theorem has now been generalised in many different directions. One of these is the multidimensional Szemerédi theorem of Furstenberg and Katznelson, who used ergodic-theoretic techniques to show that any dense subset of necessarily contained infinitely many constellations of any prescribed shape. Our main result is to relativise that theorem to the primes as well:
Theorem 2 (Multidimensional Szemerédi theorem in the primes) Let
, and let
be a subset of the
Cartesian power
of the primes of positive relative density, thus
Then for any
,
contains infinitely many “constellations” of the form
with
and
a positive integer.
In the case when is itself a Cartesian product of one-dimensional sets (in particular, if
is all of
), this result already follows from Theorem 1, but there does not seem to be a similarly easy argument to deduce the general case of Theorem 2 from previous results. Simultaneously with this paper, an independent proof of Theorem 2 using a somewhat different method has been established by Cook, Maygar, and Titichetrakun.
The result is reminiscent of an earlier result of mine on finding constellations in the Gaussian primes (or dense subsets thereof). That paper followed closely the arguments of my original paper with Ben Green, namely it first enclosed (a W-tricked version of) the primes or Gaussian primes (in a sieve theoretic-sense) by a slightly larger set (or more precisely, a weight function ) of almost primes or almost Gaussian primes, which one could then verify (using methods closely related to the sieve-theoretic methods in the ongoing Polymath8 project) to obey certain pseudorandomness conditions, known as the linear forms condition and the correlation condition. Very roughly speaking, these conditions assert statements of the following form: if
is a randomly selected integer, then the events of
simultaneously being an almost prime (or almost Gaussian prime) are approximately independent for most choices of
. Once these conditions are satisfied, one can then run a transference argument (initially based on ergodic-theory methods, but nowadays there are simpler transference results based on the Hahn-Banach theorem, due to Gowers and Reingold-Trevisan-Tulsiani-Vadhan) to obtain relative Szemerédi-type theorems from their absolute counterparts.
However, when one tries to adapt these arguments to sets such as , a new difficulty occurs: the natural analogue of the almost primes would be the Cartesian square
of the almost primes – pairs
whose entries are both almost primes. (Actually, for technical reasons, one does not work directly with a set of almost primes, but would instead work with a weight function such as
that is concentrated on a set such as
, but let me ignore this distinction for now.) However, this set
does not enjoy as many pseudorandomness conditions as one would need for a direct application of the transference strategy to work. More specifically, given any fixed
, and random
, the four events
do not behave independently (as they would if were replaced for instance by the Gaussian almost primes), because any three of these events imply the fourth. This blocks the transference strategy for constellations which contain some right-angles to them (e.g. constellations of the form
) as such constellations soon turn into rectangles such as the one above after applying Cauchy-Schwarz a few times. (But a few years ago, Cook and Magyar showed that if one restricted attention to constellations which were in general position in the sense that any coordinate hyperplane contained at most one element in the constellation, then this obstruction does not occur and one can establish Theorem 2 in this case through the transference argument.) It’s worth noting that very recently, Conlon, Fox, and Zhao have succeeded in removing of the pseudorandomness conditions (namely the correlation condition) from the transference principle, leaving only the linear forms condition as the remaining pseudorandomness condition to be verified, but unfortunately this does not completely solve the above problem because the linear forms condition also fails for
(or for weights concentrated on
) when applied to rectangular patterns.
There are now two ways known to get around this problem and establish Theorem 2 in full generality. The approach of Cook, Magyar, and Titichetrakun proceeds by starting with one of the known proofs of the multidimensional Szemerédi theorem – namely, the proof that proceeds through hypergraph regularity and hypergraph removal – and attach pseudorandom weights directly within the proof itself, rather than trying to add the weights to the result of that proof through a transference argument. (A key technical issue is that weights have to be added to all the levels of the hypergraph – not just the vertices and top-order edges – in order to circumvent the failure of naive pseudorandomness.) As one has to modify the entire proof of the multidimensional Szemerédi theorem, rather than use that theorem as a black box, the Cook-Magyar-Titichetrakun argument is lengthier than ours; on the other hand, it is more general and does not rely on some difficult theorems about primes that are used in our paper.
In our approach, we continue to use the multidimensional Szemerédi theorem (or more precisely, the equivalent theorem of Furstenberg and Katznelson concerning multiple recurrence for commuting shifts) as a black box. The difference is that instead of using a transference principle to connect the relative multidimensional Szemerédi theorem we need to the multiple recurrence theorem, we instead proceed by a version of the Furstenberg correspondence principle, similar to the one that connects the absolute multidimensional Szemerédi theorem to the multiple recurrence theorem. I had discovered this approach many years ago in an unpublished note, but had abandoned it because it required an infinite number of linear forms conditions (in contrast to the transference technique, which only needed a finite number of linear forms conditions and (until the recent work of Conlon-Fox-Zhao) a correlation condition). The reason for this infinite number of conditions is that the correspondence principle has to build a probability measure on an entire -algebra; for this, it is not enough to specify the measure
of a single set such as
, but one also has to specify the measure
of “cylinder sets” such as
where
could be arbitrarily large. The larger
gets, the more linear forms conditions one needs to keep the correspondence under control.
With the sieve weights we were using at the time, standard sieve theory methods could indeed provide a finite number of linear forms conditions, but not an infinite number, so my idea was abandoned. However, with my later work with Green and Ziegler on linear equations in primes (and related work on the Mobius-nilsequences conjecture and the inverse conjecture on the Gowers norm), Tamar and I realised that the primes themselves obey an infinite number of linear forms conditions, so one can basically use the primes (or a proxy for the primes, such as the von Mangoldt function
) as the enveloping sieve weight, rather than a classical sieve. Thus my old idea of using the Furstenberg correspondence principle to transfer Szemerédi-type theorems to the primes could actually be realised. In the one-dimensional case, this simply produces a much more complicated proof of Theorem 1 than the existing one; but it turns out that the argument works as well in higher dimensions and yields Theorem 2 relatively painlessly, except for the fact that it needs the results on linear equations in primes, the known proofs of which are extremely lengthy (and also require some of the transference machinery mentioned earlier). The problem of correlations in rectangles is avoided in the correspondence principle approach because one can compensate for such correlations by performing a suitable weighted limit to compute the measure
of cylinder sets, with each
requiring a different weighted correction. (This may be related to the Cook-Magyar-Titichetrakun strategy of weighting all of the facets of the hypergraph in order to recover pseudorandomness, although our contexts are rather different.)
Ben Green and I have just uploaded to the arXiv our paper “New bounds for Szemeredi’s theorem, Ia: Progressions of length 4 in finite field geometries revisited“, submitted to Proc. Lond. Math. Soc.. This is both an erratum to, and a replacement for, our previous paper “New bounds for Szemeredi’s theorem. I. Progressions of length 4 in finite field geometries“. The main objective in both papers is to bound the quantity for a vector space
over a finite field
of characteristic greater than
, where
is defined as the cardinality of the largest subset of
that does not contain an arithmetic progression of length
. In our earlier paper, we gave two arguments that bounded
in the regime when the field
was fixed and
was large. The first “cheap” argument gave the bound
and the more complicated “expensive” argument gave the improvement
for some constant depending only on
.
Unfortunately, while the cheap argument is correct, we discovered a subtle but serious gap in our expensive argument in the original paper. Roughly speaking, the strategy in that argument is to employ the density increment method: one begins with a large subset of
that has no arithmetic progressions of length
, and seeks to locate a subspace on which
has a significantly increased density. Then, by using a “Koopman-von Neumann theorem”, ultimately based on an iteration of the inverse
theorem of Ben and myself (and also independently by Samorodnitsky), one approximates
by a “quadratically structured” function
, which is (locally) a combination of a bounded number of quadratic phase functions, which one can prepare to be in a certain “locally equidistributed” or “locally high rank” form. (It is this reduction to the high rank case that distinguishes the “expensive” argument from the “cheap” one.) Because
has no progressions of length
, the count of progressions of length
weighted by
will also be small; by combining this with the theory of equidistribution of quadratic phase functions, one can then conclude that there will be a subspace on which
has increased density.
The error in the paper was to conclude from this that the original function also had increased density on the same subspace; it turns out that the manner in which
approximates
is not strong enough to deduce this latter conclusion from the former. (One can strengthen the nature of approximation until one restores such a conclusion, but only at the price of deteriorating the quantitative bounds on
one gets at the end of the day to be worse than the cheap argument.)
After trying unsuccessfully to repair this error, we eventually found an alternate argument, based on earlier papers of ourselves and of Bergelson-Host-Kra, that avoided the density increment method entirely and ended up giving a simpler proof of a stronger result than (1), and also gives the explicit value of for the exponent
in (1). In fact, it gives the following stronger result:
Theorem 1 Let
be a subset of
of density at least
, and let
. Then there is a subspace
of
of codimension
such that the number of (possibly degenerate) progressions
in
is at least
.
The bound (1) is an easy consequence of this theorem after choosing and removing the degenerate progressions from the conclusion of the theorem.
The main new idea is to work with a local Koopman-von Neumann theorem rather than a global one, trading a relatively weak global approximation to with a significantly stronger local approximation to
on a subspace
. This is somewhat analogous to how sometimes in graph theory it is more efficient (from the point of view of quantative estimates) to work with a local version of the Szemerédi regularity lemma which gives just a single regular pair of cells, rather than attempting to regularise almost all of the cells. This local approach is well adapted to the inverse
theorem we use (which also has this local aspect), and also makes the reduction to the high rank case much cleaner. At the end of the day, one ends up with a fairly large subspace
on which
is quite dense (of density
) and which can be well approximated by a “pure quadratic” object, namely a function of a small number of quadratic phases obeying a high rank condition. One can then exploit a special positivity property of the count of length four progressions weighted by pure quadratic objects, essentially due to Bergelson-Host-Kra, which then gives the required lower bound.
A few days ago, Endre Szemerédi was awarded the 2012 Abel prize “for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.” The full citation for the prize may be found here, and the written notes for a talk given by Tim Gowers on Endre’s work at the announcement may be found here (and video of the talk can be found here).
As I was on the Abel prize committee this year, I won’t comment further on the prize, but will instead focus on what is arguably Endre’s most well known result, namely Szemerédi’s theorem on arithmetic progressions:
Theorem 1 (Szemerédi’s theorem) Let
be a set of integers of positive upper density, thus
, where
. Then
contains an arithmetic progression of length
for any
.
Szemerédi’s original proof of this theorem is a remarkably intricate piece of combinatorial reasoning. Most proofs of theorems in mathematics – even long and difficult ones – generally come with a reasonably compact “high-level” overview, in which the proof is (conceptually, at least) broken down into simpler pieces. There may well be technical difficulties in formulating and then proving each of the component pieces, and then in fitting the pieces together, but usually the “big picture” is reasonably clear. To give just one example, the overall strategy of Perelman’s proof of the Poincaré conjecture can be briefly summarised as follows: to show that a simply connected three-dimensional manifold is homeomorphic to a sphere, place a Riemannian metric on it and perform Ricci flow, excising any singularities that arise by surgery, until the entire manifold becomes extinct. By reversing the flow and analysing the surgeries performed, obtain enough control on the topology of the original manifold to establish that it is a topological sphere.
In contrast, the pieces of Szemerédi’s proof are highly interlocking, particularly with regard to all the epsilon-type parameters involved; it takes quite a bit of notational setup and foundational lemmas before the key steps of the proof can even be stated, let alone proved. Szemerédi’s original paper contains a logical diagram of the proof (reproduced in Gowers’ recent talk) which already gives a fair indication of this interlocking structure. (Many years ago I tried to present the proof, but I was unable to find much of a simplification, and my exposition is probably not that much clearer than the original text.) Even the use of nonstandard analysis, which is often helpful in cleaning up armies of epsilons, turns out to be a bit tricky to apply here. (In typical applications of nonstandard analysis, one can get by with a single nonstandard universe, constructed as an ultrapower of the standard universe; but to correctly model all the epsilons occuring in Szemerédi’s argument, one needs to repeatedly perform the ultrapower construction to obtain a (finite) sequence of increasingly nonstandard (and increasingly saturated) universes, each one containing unbounded quantities that are far larger than any quantity that appears in the preceding universe, as discussed at the end of this previous blog post. This sequence of universes does end up concealing all the epsilons, but it is not so clear that this is a net gain in clarity for the proof; I may return to the nonstandard presentation of Szemeredi’s argument at some future juncture.)
Instead of trying to describe the entire argument here, I thought I would instead show some key components of it, with only the slightest hint as to how to assemble the components together to form the whole proof. In particular, I would like to show how two particular ingredients in the proof – namely van der Waerden’s theorem and the Szemerédi regularity lemma – become useful. For reasons that will hopefully become clearer later, it is convenient not only to work with ordinary progressions , but also progressions of progressions
, progressions of progressions of progressions, and so forth. (In additive combinatorics, these objects are known as generalised arithmetic progressions of rank one, two, three, etc., and play a central role in the subject, although the way they are used in Szemerédi’s proof is somewhat different from the way that they are normally used in additive combinatorics.) Very roughly speaking, Szemerédi’s proof begins by building an enormous generalised arithmetic progression of high rank containing many elements of the set
(arranged in a “near-maximal-density” configuration), and then steadily prunes this progression to improve the combinatorial properties of the configuration, until one ends up with a single rank one progression of length
that consists entirely of elements of
.
To illustrate some of the basic ideas, let us first consider a situation in which we have located a progression of progressions of length
, with each progression
,
being quite long, and containing a near-maximal amount of elements of
, thus
where is the “maximal density” of
along arithmetic progressions. (There are a lot of subtleties in the argument about exactly how good the error terms are in various approximations, but we will ignore these issues for the sake of this discussion and just use the imprecise symbols such as
instead.) By hypothesis,
is positive. The objective is then to locate a progression
in
, with each
in
for
. It may help to view the progression of progressions
as a tall thin rectangle
.
If we write for
, then the problem is equivalent to finding a (possibly degenerate) arithmetic progression
, with each
in
.
By hypothesis, we know already that each set has density about
in
:
Let us now make a “weakly mixing” assumption on the , which roughly speaking asserts that
for “most” subsets of
of density
of a certain form to be specified shortly. This is a plausible type of assumption if one believes
to behave like a random set, and if the sets
are constructed “independently” of the
in some sense. Of course, we do not expect such an assumption to be valid all of the time, but we will postpone consideration of this point until later. Let us now see how this sort of weakly mixing hypothesis could help one count progressions
of the desired form.
We will inductively consider the following (nonrigorously defined) sequence of claims for each
:
-
: For most choices of
, there are
arithmetic progressions
in
with the specified choice of
, such that
for all
.
(Actually, to avoid boundary issues one should restrict to lie in the middle third of
, rather than near the edges, but let us ignore this minor technical detail.) The quantity
is natural here, given that there are
arithmetic progressions
in
that pass through
in the
position, and that each one ought to have a probability of
or so that the events
simultaneously hold.) If one has the claim
, then by selecting a typical
in
, we obtain a progression
with
for all
, as required. (In fact, we obtain about
such progressions by this method.)
We can heuristically justify the claims by induction on
. For
, the claims
are clear just from direct counting of progressions (as long as we keep
away from the edges of
). Now suppose that
, and the claims
have already been proven. For any
and for most
, we have from hypothesis that there are
progressions
in
through
with
. Let
be the set of all the values of
attained by these progressions, then
. Invoking the weak mixing hypothesis, we (heuristically, at least) conclude that for most choices of
, we have
which then gives the desired claim .
The observant reader will note that we only needed the claim in the case
for the above argument, but for technical reasons, the full proof requires one to work with more general values of
(also the claim
needs to be replaced by a more complicated version of itself, but let’s ignore this for sake of discussion).
We now return to the question of how to justify the weak mixing hypothesis (2). For a single block of
, one can easily concoct a scenario in which this hypothesis fails, by choosing
to overlap with
too strongly, or to be too disjoint from
. However, one can do better if one can select
from a long progression of blocks. The starting point is the following simple double counting observation that gives the right upper bound:
Proposition 2 (Single upper bound) Let
be a progression of progressions
for some large
. Suppose that for each
, the set
has density
in
(i.e. (1) holds). Let
be a subset of
of density
. Then (if
is large enough) one can find an
such that
Proof: The key is the double counting identity
Because has maximal density
and
is large, we have
for each , and thus
The claim then follows from the pigeonhole principle.
Now suppose we want to obtain weak mixing not just for a single set , but for a small number
of such sets, i.e. we wish to find an
for which
for all , where
is the density of
in
. The above proposition gives, for each
, a choice of
for which (3) holds, but it could be a different
for each
, and so it is not immediately obvious how to use Proposition 2 to find an
for which (3) holds simultaneously for all
. However, it turns out that the van der Waerden theorem is the perfect tool for this amplification:
Proposition 3 (Multiple upper bound) Let
be a progression of progressions
for some large
. Suppose that for each
, the set
has density
in
(i.e. (1) holds). For each
, let
be a subset of
of density
. Then (if
is large enough depending on
) one can find an
such that
simultaneously for all
.
Proof: Suppose that the claim failed (for some suitably large ). Then, for each
, there exists
such that
This can be viewed as a colouring of the interval by
colours. If we take
large compared to
, van der Waerden’s theorem allows us to then find a long subprogression of
which is monochromatic, so that
is constant on this progression. But then this will furnish a counterexample to Proposition 2.
One nice thing about this proposition is that the upper bounds can be automatically upgraded to an asymptotic:
Proposition 4 (Multiple mixing) Let
be a progression of progressions
for some large
. Suppose that for each
, the set
has density
in
(i.e. (1) holds). For each
, let
be a subset of
of density
. Then (if
is large enough depending on
) one can find an
such that
simultaneously for all
.
Proof: By applying the previous proposition to the collection of sets and their complements
(thus replacing
with
, one can find an
for which
and
which gives the claim.
However, this improvement of Proposition 2 turns out to not be strong enough for applications. The reason is that the number of sets
for which mixing is established is too small compared with the length
of the progression one has to use in order to obtain that mixing. However, thanks to the magic of the Szemerédi regularity lemma, one can amplify the above proposition even further, to allow for a huge number of
to be mixed (at the cost of excluding a small fraction of exceptions):
Proposition 5 (Really multiple mixing) Let
be a progression of progressions
for some large
. Suppose that for each
, the set
has density
in
(i.e. (1) holds). For each
in some (large) finite set
, let
be a subset of
of density
. Then (if
is large enough, but not dependent on the size of
) one can find an
such that
simultaneously for almost all
.
Proof: We build a bipartite graph connecting the progression
to the finite set
by placing an edge
between an element
and an element
whenever
. The number
can then be interpreted as the degree of
in this graph, while the number
is the number of neighbours of
that land in
.
We now apply the regularity lemma to this graph . Roughly speaking, what this lemma does is to partition
and
into almost equally sized cells
and
such that for most pairs
of cells, the graph
resembles a random bipartite graph of some density
between these two cells. The key point is that the number
of cells here is bounded uniformly in the size of
and
. As a consequence of this lemma, one can show that for most vertices
in a typical cell
, the number
is approximately equal to
and the number is approximately equal to
The point here is that the different statistics
are now controlled by a mere
statistics
(this is not unlike the use of principal component analysis in statistics, incidentally, but that is another story). Now, we invoke Proposition 4 to find an
for which
simultaneously for all , and the claim follows.
This proposition now suggests a way forward to establish the type of mixing properties (2) needed for the preceding attempt at proving Szemerédi’s theorem to actually work. Whereas in that attempt, we were working with a single progression of progressions of progressions containing a near-maximal density of elements of
, we will now have to work with a family
of such progression of progressions, where
ranges over some suitably large parameter set. Furthermore, in order to invoke Proposition 5, this family must be “well-arranged” in some arithmetic sense; in particular, for a given
, it should be possible to find many reasonably large subfamilies of this family for which the
terms
of the progression of progressions in this subfamily are themselves in arithmetic progression. (Also, for technical reasons having to do with the fact that the sets
in Proposition 5 are not allowed to depend on
, one also needs the progressions
for any given
to be “similar” in the sense that they intersect
in the same fashion (thus the sets
as
varies need to be translates of each other).) If one has this sort of family, then Proposition 5 allows us to “spend” some of the degrees of freedom of the parameter set
in order to gain good mixing properties for at least one of the sets
in the progression of progressions.
Of course, we still have to figure out how to get such large families of well-arranged progressions of progressions. Szemerédi’s solution was to begin by working with generalised progressions of a much larger rank than the rank
progressions considered here; roughly speaking, to prove Szemerédi’s theorem for length
progressions, one has to consider generalised progressions of rank as high as
. It is possible by a reasonably straightforward (though somewhat delicate) “density increment argument” to locate a huge generalised progression of this rank which is “saturated” by
in a certain rather technical sense (related to the concept of “near maximal density” used previously). Then, by another reasonably elementary argument, it is possible to locate inside a suitable large generalised progression of some rank
, a family of large generalised progressions of rank
which inherit many of the good properties of the original generalised progression, and which have the arithmetic structure needed for Proposition 5 to be applicable, at least for one value of
. (But getting this sort of property for all values of
simultaneously is tricky, and requires many careful iterations of the above scheme; there is also the problem that by obtaining good behaviour for one index
, one may lose good behaviour at previous indices, leading to a sort of “Tower of Hanoi” situation which may help explain the exponential factor in the rank
that is ultimately needed. It is an extremely delicate argument; all the parameters and definitions have to be set very precisely in order for the argument to work at all, and it is really quite remarkable that Endre was able to see it through to the end.)
In 1977, Furstenberg established his multiple recurrence theorem:
Theorem 1 (Furstenberg multiple recurrence) Let
be a measure-preserving system, thus
is a probability space and
is a measure-preserving bijection such that
and
are both measurable. Let
be a measurable subset of
of positive measure
. Then for any
, there exists
such that
Equivalently, there exists
and
such that
As is well known, the Furstenberg multiple recurrence theorem is equivalent to Szemerédi’s theorem, thanks to the Furstenberg correspondence principle; see for instance these lecture notes of mine.
The multiple recurrence theorem is proven, roughly speaking, by an induction on the “complexity” of the system . Indeed, for very simple systems, such as periodic systems (in which
is the identity for some
, which is for instance the case for the circle shift
,
with a rational shift
), the theorem is trivial; at a slightly more advanced level, almost periodic (or compact) systems (in which
is a precompact subset of
for every
, which is for instance the case for irrational circle shifts), is also quite easy. One then shows that the multiple recurrence property is preserved under various extension operations (specifically, compact extensions, weakly mixing extensions, and limits of chains of extensions), which then gives the multiple recurrence theorem as a consequence of the Furstenberg-Zimmer structure theorem for measure-preserving systems. See these lecture notes for further discussion.
From a high-level perspective, this is still one of the most conceptual proofs known of Szemerédi’s theorem. However, the individual components of the proof are still somewhat intricate. Perhaps the most difficult step is the demonstration that the multiple recurrence property is preserved under compact extensions; see for instance these lecture notes, which is devoted entirely to this step. This step requires quite a bit of measure-theoretic and/or functional analytic machinery, such as the theory of disintegrations, relatively almost periodic functions, or Hilbert modules.
However, I recently realised that there is a special case of the compact extension step – namely that of finite extensions – which avoids almost all of these technical issues while still capturing the essence of the argument (and in particular, the key idea of using van der Waerden’s theorem). As such, this may serve as a pedagogical device for motivating this step of the proof of the multiple recurrence theorem.
Let us first explain what a finite extension is. Given a measure-preserving system , a finite set
, and a measurable map
from
to the permutation group of
, one can form the finite extension
which as a probability space is the product of with the finite probability space
(with the discrete
-algebra and uniform probability measure), and with shift map
One easily verifies that this is indeed a measure-preserving system. We refer to as the cocycle of the system.
An example of finite extensions comes from group theory. Suppose we have a short exact sequence
of finite groups. Let be a group element of
, and let
be its projection in
. Then the shift map
on
(with the discrete
-algebra and uniform probability measure) can be viewed as a finite extension of the shift map
on
(again with the discrete
-algebra and uniform probability measure), by arbitrarily selecting a section
that inverts the projection map, identifying
with
by identifying
with
for
, and using the cocycle
Thus, for instance, the unit shift on
can be thought of as a finite extension of the unit shift
on
whenever
is a multiple of
.
Another example comes from Riemannian geometry. If is a Riemannian manifold that is a finite cover of another Riemannian manifold
(with the metric on
being the pullback of that on
), then (unit time) geodesic flow on the cosphere bundle of
is a finite extension of the corresponding flow on
.
Here, then, is the finite extension special case of the compact extension step in the proof of the multiple recurrence theorem:
Proposition 2 (Finite extensions) Let
be a finite extension of a measure-preserving system
. If
obeys the conclusion of the Furstenberg multiple recurrence theorem, then so does
.
Before we prove this proposition, let us first give the combinatorial analogue.
Lemma 3 Let
be a subset of the integers that contains arbitrarily long arithmetic progressions, and let
be a colouring of
by
colours (or equivalently, a partition of
into
colour classes
). Then at least one of the
contains arbitrarily long arithmetic progressions.
Proof: By the infinite pigeonhole principle, it suffices to show that for each , one of the colour classes
contains an arithmetic progression of length
.
Let be a large integer (depending on
and
) to be chosen later. Then
contains an arithmetic progression of length
, which may be identified with
. The colouring of
then induces a colouring on
into
colour classes. Applying (the finitary form of) van der Waerden’s theorem, we conclude that if
is sufficiently large depending on
and
, then one of these colouring classes contains an arithmetic progression of length
; undoing the identification, we conclude that one of the
contains an arithmetic progression of length
, as desired.
Of course, by specialising to the case , we see that the above Lemma is in fact equivalent to van der Waerden’s theorem.
Now we prove Proposition 2.
Proof: Fix . Let
be a positive measure subset of
. By Fubini’s theorem, we have
where and
is the fibre of
at
. Since
is positive, we conclude that the set
is a positive measure subset of . Note for each
, we can find an element
such that
. While not strictly necessary for this argument, one can ensure if one wishes that the function
is measurable by totally ordering
, and then letting
the minimal element of
for which
.
Let be a large integer (which will depend on
and the cardinality
of
) to be chosen later. Because
obeys the multiple recurrence theorem, we can find a positive integer
and
such that
Now consider the sequence of points
for . From (1), we see that
for some sequence . This can be viewed as a colouring of
by
colours, where
is the cardinality of
. Applying van der Waerden’s theorem, we conclude (if
is sufficiently large depending on
and
) that there is an arithmetic progression
in
with
such that
for some . If we then let
, we see from (2) that
for all , and the claim follows.
Remark 1 The precise connection between Lemma 3 and Proposition 2 arises from the following observation: with
as in the proof of Proposition 2, and
, the set
can be partitioned into the classes
where
is the graph of
. The multiple recurrence property for
ensures that
contains arbitrarily long arithmetic progressions, and so therefore one of the
must also, which gives the multiple recurrence property for
.
Remark 2 Compact extensions can be viewed as a generalisation of finite extensions, in which the fibres are no longer finite sets, but are themselves measure spaces obeying an additional property, which roughly speaking asserts that for many functions
on the extension, the shifts
of
behave in an almost periodic fashion on most fibres, so that the orbits
become totally bounded on each fibre. This total boundedness allows one to obtain an analogue of the above colouring map
to which van der Waerden’s theorem can be applied.
In Notes 3, we saw that the number of additive patterns in a given set was (in principle, at least) controlled by the Gowers uniformity norms of functions associated to that set.
Such norms can be defined on any finite additive group (and also on some other types of domains, though we will not discuss this point here). In particular, they can be defined on the finite-dimensional vector spaces over a finite field
.
In this case, the Gowers norms are closely tied to the space
of polynomials of degree at most
. Indeed, as noted in Exercise 20 of Notes 4, a function
of
norm
has
norm equal to
if and only if
for some
; thus polynomials solve the “
inverse problem” for the trivial inequality
. They are also a crucial component of the solution to the “
inverse problem” and “
inverse problem”. For the former, we will soon show:
Proposition 1 (
inverse theorem for
) Let
be such that
and
for some
. Then there exists
such that
, where
is a constant depending only on
.
Thus, for the Gowers norm to be almost completely saturated, one must be very close to a polynomial. The converse assertion is easily established:
Exercise 1 (Converse to
inverse theorem for
) If
and
for some
, then
, where
is a constant depending only on
.
In the world, one no longer expects to be close to a polynomial. Instead, one expects to correlate with a polynomial. Indeed, one has
Lemma 2 (Converse to the
inverse theorem for
) If
and
are such that
, where
, then
.
Proof: From the definition of the norm (equation (18) from Notes 3), the monotonicity of the Gowers norms (Exercise 19 of Notes 3), and the polynomial phase modulation invariance of the Gowers norms (Exercise 21 of Notes 3), one has
and the claim follows.
In the high characteristic case at least, this can be reversed:
Theorem 3 (
inverse theorem for
) Suppose that
. If
is such that
and
, then there exists
such that
.
This result is sometimes referred to as the inverse conjecture for the Gowers norm (in high, but bounded, characteristic). For small , the claim is easy:
Exercise 2 Verify the cases
of this theorem. (Hint: to verify the
case, use the Fourier-analytic identities
and
, where
is the space of all homomorphisms
from
to
, and
are the Fourier coefficients of
.)
This conjecture for larger values of are more difficult to establish. The
case of the theorem was established by Ben Green and myself in the high characteristic case
; the low characteristic case
was independently and simultaneously established by Samorodnitsky. The cases
in the high characteristic case was established in two stages, firstly using a modification of the Furstenberg correspondence principle, due to Ziegler and myself. to convert the problem to an ergodic theory counterpart, and then using a modification of the methods of Host-Kra and Ziegler to solve that counterpart, as done in this paper of Bergelson, Ziegler, and myself.
The situation with the low characteristic case in general is still unclear. In the high characteristic case, we saw from Notes 4 that one could replace the space of non-classical polynomials in the above conjecture with the essentially equivalent space of classical polynomials
. However, as we shall see below, this turns out not to be the case in certain low characteristic cases (a fact first observed by Lovett, Meshulam, and Samorodnitsky, and independently by Ben Green and myself), for instance if
and
; this is ultimately due to the existence in those cases of non-classical polynomials which exhibit no significant correlation with classical polynomials of equal or lesser degree. This distinction between classical and non-classical polynomials appears to be a rather non-trivial obstruction to understanding the low characteristic setting; it may be necessary to obtain a more complete theory of non-classical polynomials in order to fully settle this issue.
The inverse conjecture has a number of consequences. For instance, it can be used to establish the analogue of Szemerédi’s theorem in this setting:
Theorem 4 (Szemerédi’s theorem for finite fields) Let
be a finite field, let
, and let
be such that
. If
is sufficiently large depending on
, then
contains an (affine) line
for some
with
.
Exercise 3 Use Theorem 4 to establish the following generalisation: with the notation as above, if
and
is sufficiently large depending on
, then
contains an affine
-dimensional subspace.
We will prove this theorem in two different ways, one using a density increment method, and the other using an energy increment method. We discuss some other applications below the fold.
Ben Green, and I have just uploaded to the arXiv a short (six-page) paper “Yet another proof of Szemeredi’s theorem“, submitted to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we put in print a folklore observation, namely that the inverse conjecture for the Gowers norm, together with the density increment argument, easily implies Szemerédi’s famous theorem on arithmetic progressions. This is unsurprising, given that Gowers’ proof of Szemerédi’s theorem proceeds through a weaker version of the inverse conjecture and a density increment argument, and also given that it is possible to derive Szemerédi’s theorem from knowledge of the characteristic factor for multiple recurrence (the ergodic theory analogue of the inverse conjecture, first established by Host and Kra), as was done by Bergelson, Leibman, and Lesigne (and also implicitly in the earlier paper of Bergelson, Host, and Kra); but to our knowledge the exact derivation of Szemerédi’s theorem from the inverse conjecture was not in the literature. Ordinarily this type of folklore might be considered too trifling (and too well known among experts in the field) to publish; but we felt that the venue of the Szemerédi birthday conference provided a natural venue for this particular observation.
The key point is that one can show (by an elementary argument relying primarily an induction on dimension argument and the Weyl recurrence theorem, i.e. that given any real and any integer
, that the expression
gets arbitrarily close to an integer) that given a (polynomial) nilsequence
, one can subdivide any long arithmetic progression (such as
) into a number of medium-sized progressions, where the nilsequence is nearly constant on each progression. As a consequence of this and the inverse conjecture for the Gowers norm, if a set has no arithmetic progressions, then it must have an elevated density on a subprogression; iterating this observation as per the usual density-increment argument as introduced long ago by Roth, one obtains the claim. (This is very close to the scheme of Gowers’ proof.)
Technically, one might call this the shortest proof of Szemerédi’s theorem in the literature (and would be something like the sixteenth such genuinely distinct proof, by our count), but that would be cheating quite a bit, primarily due to the fact that it assumes the inverse conjecture for the Gowers norm, our current proof of which is checking in at about 100 pages…
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