The construction of a set of recurrence which is not a set of strong recurrence

Abstract

This paper deals with two possible definitions of recurrence in measure preserving systems. A set of integersR is said to be a set of (Poincare) recurrence if, for all measure preserving systems (X, B, μ, T) and any measurable setA of positive measure, there is anr eR such thatμ(T r A∩A)>0.R is said to be a set of strong recurrence if, for all measure preserving systems (X, B, μ, T) and any measurable setA of positive measure, there is ane>0 and an infinite number of elementsr ofR such thatμ(T r A∩A)≥e (see Bergelson’s 1985 paper). This paper constructs a set of recurrenceR, an example of a measure preserving system (X, B, μ, T) and a measurable setA of measure 1/2, such that lim r→∞:reRμ (A∩T r A)=0. In particularR is a set of recurrence but not a set of strong recurrence, giving a negative answer to a question of Bergelson posed in 1985. Further, it also constructs a set of recurrence which does not force the continuity of positive measures and so reproves a result of Bourgain published in 1987.