Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory
type: Item
instance of: scholarly article
title: Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory
zbMATH Open document ID: 1146.37007
author: V. Bergelson
author: A. Leibman
author: Emmanuel Lesigne
published in: Journal d'Analyse Mathématique
publication date: 2008-05-27T00:00:00Z
review text: After introducing two notions of complexity of a system of polynomials \(p_1,\dots,p_r\in\mathbb{Z}[n]\), the authors characterize the limits of the expressions of the form \(\mu(A_0\cap T^{-p_1(n)}A_1\cap\dots\cap T^{-p_r(n)}A_r)\), where \(T\) is a skew-product transformation of a torus \(\mathbb{T}^d\) and \(A_i\subseteq\mathbb{T}^d\) are measurable sets. Using the dynamical results they obtain, they are able to build subsets of integers with specific combinatorial properties related with the polynomial Szemerédi theorem.
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 5279116
zbMATH Keywords: complexity of a system of polynomials
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: polynomial Szemerédi theorem
zbMATH Keywords: measure-preserving transformations
MaRDI profile type: Publication
full work available at URL: s11854-008-0002-z
OpenAlex ID: W1994747108
cites work: Weakly mixing PET
cites work: Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
cites work: Q4880550
cites work: Multiple ergodic averages for three polynomials and applications
cites work: Polynomial averages converge to the product of integrals
cites work: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions
cites work: Q3909268
cites work: Q3496434
cites work: Nonconventional ergodic averages and nilmanifolds
cites work: Convergence of polynomial ergodic averages
cites work: Convergence of multiple ergodic averages along polynomials of several variables
cites work: On sets of integers containing k elements in arithmetic progression
cites work: Universal characteristic factors and Furstenberg averages
DOI: 10.1007/S11854-008-0002-Z
description: scientific article; zbMATH DE number 5279116