Multiple recurrence and nilsequences

• 成果类型: Article
• 署名作者: Bergelson, V; Host, B; Kra, B; Ruzsa, I
• 署名单位: University System of Ohio; Ohio State University; Universite Gustave-Eiffel; Northwestern University; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Hungarian Academy of Sciences
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0428-6
• 发表日期: 2005
• 页码: 261-303

摘要

Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving system (X, X, mu, T) and a set A is an element of X of positive measure, the set of integers n such that mu( A boolean AND T-n A boolean AND T-2n A boolean AND... boolean AND T-kn A) >mu( A)(k+1) - epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k + 1) of the arithmetic progression under consideration. In an ergodic system, for k = 2 and k = 3, this set is syndetic, while for k >= 4 it is not. The main tool is a decomposition result for the multicorrelation sequence integral f( x) f(T(n)x) f(T(2n)x)... f(T(kn)x) d mu( x), where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d*( E) > 0 and for all epsilon > 0, the set {n is an element of Z: d* (E boolean AND ( E + n) boolean AND ( E + 2n) boolean AND ( E + 3n)) > d*( E)(4) - epsilon} is syndetic.