Related papers: Ergodic Methods in Additive Combinatorics

We establish an abstract local ergodic theorem, under suitable space-time scaling, for the (boundary-driven) symmetric exclusion process on an increasing sequence of balls covering an infinite weighted graph. The proofs are based on 1-block…

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…

We prove structural results for measure preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions these systems always have rational…

In this paper we study ergodic $\mathbb{Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions there are always directions which expand significantly a given measurable set…

In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the…

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds…

We prove the existence of a successful coupling for $n$ particles in the symmetric inclusion process. As a consequence we characterize the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in…

Furstenberg, Glasscock, Bergelson, Beiglboeck have been studied abundance in arithmatic progression on various large sets like piecewise syndetic, central, thick, etc. but also there are so many sets in which abundance in progression is…

A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity…

Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for…

We introduce the notion of common conditional expectation to investigate Birkhoff's ergodic theorem and subadditive ergodic theorem for invariant upper probabilities. If in addition, the upper probability is ergodic, we construct an…

We introduce a new class of sparse sequences that are ergodic and pointwise universally $L^2$-good for ergodic averages. That is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions.…

Bergelson et al. observed that Furstenberg's proof of Szemeredi's theorem provides a positive lower bound on the density of arithmetic progressions in sets of positive density in the integers. Namely, for every $\delta\in(0,1]$ and every…

The objective of this paper is to characterize the structure of the set $\Theta$ for a continuous ergodic upper probability $\mathbb{V}=\sup_{P\in\Theta}P$ (Theorem \ref {main result}): . $\Theta$ contains a finite number of ergodic…

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…

Combining Freiman's theorem with Balog-Szemeredi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In…

Motivated by studying stochastic systems with non-Gaussian L\'evy noise, spectral properties for a type of linear cocycles are considered. These linear cocycles have countable jump discontinuities in time. A multiplicative ergodic theorem…

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as…

Conical density theorems are used in the geometric measure theory to derive geometric information from given metric information. The idea is to examine how a measure is distributed in small balls. Finding conditions that guarantee the…