A famous theorem of Szemerédi asserts that given any density $0 < δ\leq 1$ and any integer $k \geq 3$, any set of integers with density $δ$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemerédi's original combinatorial proof using the Szemerédi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers' more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is ``elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.
By using ergodic theoretic techniques following Hillel Fürstenberg, we prove that measurable subsets of a locally compact abelian group of positive upper density contain Szemerédi-wise configurations defined by an arbitrary compact subset of the group.
A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. …
Szemerédi's Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's Theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from t…
In 1975 Szemerédi proved the long-standing conjecture of Erdős and Turán that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemerédi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemerédi's Theorem by first showing its equivalence to an ergodic-theoretic assertion of multiple recurrence, and then bringing new machinery in ergodic theory to bear on proving that. His ergodic-theoretic approach subsequently yielded several other results in extremal combinatorics, as well as revea…
We exhibit proofs of two ergodic-theoretic results in the study of multiple recurrence using an analog of the density-increment argument of Roth and Gowers: Furstenberg's Multiple Recurrence Theorem (which implies Szemerédi's Theorem), and a two-dimensional special case of Furstenberg and Katznelson's multidimensional version of this theorem. The second of these requires also an analog of some recent finitary work by Shkredov. Many proofs of these multiple recurrence theorems are now known, but our main goal is to shed some further light on the heuristic correspondence principle that has grown…
We present a short proof of Szemerédi's Theorem using a dynamical system enriched by ideas from model theory. The resulting proof contains features reminiscent of proofs based on both ergodic theory and on hypergraph regularity.
We obtain quantitative bounds in the polynomial Szemerédi theorem of Bergelson and Leibman, provided the polynomials are homogeneous and of the same degree. Such configurations include arithmetic progressions with common difference equal to a perfect kth power.
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemerédi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbol…
A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge in the mean to the product of the integrals. We give necessary and sufficient conditions for joint ergodicity that are flexible enough to recover most of the known examples of jointly ergodic sequences and also allow us to answer some related open problems. An interesting feature of our arguments is that they avoid deep tools from ergodic theory that were previously used to establish similar results. Our…
In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form $p(n)$ where $p(n)$ is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem can be of the form $[n^δ]$ where $δ$ is any positive real number and $[x]$ denotes …
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter to those integers that have an even number of distinct prime factors, or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are the…
Let $P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}]$ be a family of polynomials such that $p_{i}(\Z^{m})\sle\Z$, $i=1,\ld,r$. We say that the family $P$ has {\it PSZ property} if for any set $E\sle\Z$ with $d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0$ there exist infinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression of the form \hbox{$\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}$}. We prove that a polynomial family $P=\{p_{1},\ld,p_{r}\}$ has PSZ property if and only if the polynomials $p_{1},\ld,p_{r}$ are {\it jointly intersective}, meaning that for any $k\in\N$ t…
We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form $\{l_1p,l_2p,...,l_kp\}$. We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that …
We deduce, as a consequence of the arithmetic removal lemma, an almost-all version of the Balog-Szemerédi-Gowers theorem: For any $K\geq 1$ and $\varepsilon > 0$, there exists $δ= δ(K,\varepsilon)>0$ such that the following statement holds: if $|A+_ΓA| \leq K|A|$ for some $Γ\geq (1-δ)|A|^2$, then there is a subset $A' \subset A$ with $|A'| \geq (1-\varepsilon)|A|$ such that $|A'+A'| \leq |A+_ΓA| + \varepsilon |A|$. We also discuss issues around quantitative bounds in this statement, in particular showing that when $A \subset \mathbb{Z}$ the dependence of $δ$ on $ε$ cannot be polynomial for any…
Recently Conlon, Fox, and the author gave a new proof of a relative Szemerédi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemerédi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition. This note contains an alternative proof of the new rel…
We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K^{…
Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a genera…
In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemerédi's theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs-de Leeuw-Glicksberg decomposition and properties of the Gowers-Host-Kra uniformity seminorms.
The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-colouring of the n-dimensional grid {1,...,k}^n contains a combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdos and Turan in 1936, proved by Szemeredi in 1975, and given a different proof by Furstenberg in 1977. The Hales-Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a signific…
We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit…
The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemerédi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set $A + A$ small and then deriving that the product set $AA$ is large (using Freiman's structure theorem). (cf [N-T], [Na3].) We follow the reverse route and prove that if $|AA| < c|A|$, then …
Extending Furstenberg's ergodic theoretic proof for Szemerédi's theorem on arithmetic progressions, Furstenberg and Weiss (2003) proved the following qualitative result. For every d and k, there exists an integer N such that no matter how we color the vertices of a complete binary tree T_N of depth N with k colors, we can find a monochromatic replica of T_d in T_N such that (1) all vertices at the same level in T_d are mapped into vertices at the same level in T_N; (2) if a vertex x of T_d is mapped into a vertex y in T_N, then the two children of x are mapped into descendants of the the two c…
We show that there exists $c>0$ such that any subset of $\{1, \dots, N\}$ of density at least $(\log\log{N})^{-c}$ contains a nontrivial progression of the form $x,x+y,x+y^2$. This is the first quantitatively effective version of the Bergelson--Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations $x,x+y,x+y^2$ deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and…
(This text is a survey written for the Bourbaki seminar on the work of F. Manners.) Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemerédi's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler…
