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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
- V. Bergelson, B. Host, Bryna Kra, I. Ruzsa
- Mathematics
- 1 March 2005
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
- B. Host
- Mathematics
- 22 November 2008
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
- F. Durand, B. Host, Christian F. Skau
- MathematicsErgodic Theory and Dynamical Systems
- 1 August 1999
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
- F. Blanchard, B. Host, Sylvie Ruette
- MathematicsErgodic Theory and Dynamical Systems
- 1 June 2002
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
- N. Frantzikinakis, B. Host
- Mathematics
- 4 March 2014
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
- N. Frantzikinakis, B. Host, Bryna Kra
- Mathematics
- 25 July 2006
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…
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