On uniformly distributed sequences of integers and Poincaré recurrence II

Abstract

Suppose k = ( k n ) ∞ n = 1 is a sequence of natural numbers which together with being Hartman uniform distributed satisfies certain growth conditions specified below as condition H. We introduce two notions of density on N. Given a subset E of N, we say d ∗(E) = limsup n → ∞ ¦E ∩ [1,n]¦ n , denotes its upper density. If the above limit exists we say E has density denoted d( E). We say a set S contained in N has positive Banach density of there exists a collection of finite subintervals I = ( I n ∞ n = 1 of N whose lengths tend to infinity with n such that B(S, I) = limsup n → ∞ ¦S∩I n¦ ¦I n¦ > 0 . Let B( E) = sup I B( E, I) where the supremum is taken over all such collections I. We refer to B( S) as the Banach density of S. In this paper we prove the following theorem: Suppose the subset E of N has positive Banach density B( E). Then if ( k n ) ∞ n = 1 satisfies condition H there exists a subset R of N with d( R) ≥ B( E) such that for each finite subset { n 1,…, n r } of R we have B( E ∩ ( E + k n 1 ) ∩…∩ ( E + k n r )) > 0.