Related papers: Ergodic Theorem involving additive and multiplicat…

We establish a polynomial ergodic theorem for actions of the affine group of a countable field $K$. As an application, we deduce--via a variant of Furstenberg's correspondence principle--that for fields of characteristic zero, any "large"…

Shortly after Szemer\'edi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of ergodic Ramsey…

We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-S\'{a}rk\"{o}zy theorem on squares in sets of…

We show that if $E \subset \mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements, and $|E| \geq \rho q^d$, where $ q^{-\frac{1}{2}}\ll \rho \leq 1$, then $E$ contains an isometric copy of at least $c…

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 Szemer\'edi's Theorem),…

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between…

We provide a framework for studying randomly coloured point sets in a locally compact, second-countable space on which a metrisable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical…

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach,…

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev…

We show that the equational theory of the structure $\langle \omega^{\omega}: (x,y)\mapsto x+y, x\mapsto \omega x \rangle $ is finitely axiomatizable and give a simple axiom schema when the domain is the set of transfinite ordinals. We give…

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in…

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain…

An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More…

We show any subset $A\subset\mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n\alpha],\dots,m+k[n\alpha]\}$, for some $m\in\mathbb{N}$ and $n=p-1$ for some prime $p$, where…

For two countably infinite fields whose multiplicative groups are isomorphic, we examine invariant couplings between the actions that these groups induce on the additive Pontryagin duals of the fields. We show that the actions are disjoint…

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms…

A famous theorem of Szemer\'edi 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…

Ergodic and combinatorial results obtained in [10] involved measure preserving actions of the affine group ${\mathcal A}_K$ of a countable field $K$. In this paper we develop a new approach based on ultrafilter limits which allows one to…