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Nonconventional ergodic averages and nilmanifolds
We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions
Multiple recurrence and nilsequences
Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system $(X,\mathcal{X},\mu,T)$ and a set
Ergodic seminorms for commuting transformations and applications
Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although
Substitutional dynamical systems, Bratteli diagrams and dimension groups
We explore substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit and render an
Asymptotic pairs in positive-entropy systems
We show that in a topological dynamical system (X,T) of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs (x,y) such that x\not= y and \lim_{n\to +\infty} d(T^n x,T^n
Higher order Fourier analysis of multiplicative functions and applications
We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and
Uniformity seminorms on ℓ∞ and applications
A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by
Parallelepipeds, nilpotent groups and Gowers norms
Dans sa preuve du theoreme de Szemeredi, Gowers a introduit certaines normes definies par sommation sur des parallelepipedes. Il est naturel de se demander sous quelles hypotheses on peut generaliser
Multiple recurrence and convergence for sequences related to the prime numbers
For any measure preserving system (X, , μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show the