Treffer: Ensembles intersectifs et récurrence de Poincaré. (Intersective sets and Poincaré recurrence)

Let (X,B,\(\mu)\) be a probability space and \(T: X\to X\) a measure preserving mapping; (X,B,\(\mu\),T) is called a dynamical system. A subset P of the positive integers \({\mathbb{N}}\) is called Poincaré set if there exist a dynamical system and a set A of positive measure such that there exists \(m\in P\) with \(\mu (T^{-m}A\cap A)\geq 0.\) A subset H of \({\mathbb{N}}\) is called intersective set if for every \(S\subseteq {\mathbb{N}}\) of positive density \(H\cap (S-S)\neq \emptyset,\) where \(S-S=\{a-b: a,b\in S,\quad a\neq b\}.\) It is proved that Poincaré sets and intersective sets are equal; cf. \textit{I. Z. Ruzsa} [Stud. Sci. Math. Hung. 13, 319-326 (1978; Zbl 0423.10027); Sémin. Théor. Nombres, Univ. Bordeaux I, 1982/83, Exp. No.20, No.20 bis (1973; Zbl 0529.10046, Zbl 0529.10048)], \textit{T. Kamae} and \textit{M. Mendès France} [Isr. J. Math. 31, 335-342 (1978; Zbl 0396.10040)] and \textit{H. Furstenberg} [Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press (1981; Zbl 0459.28023)]. The results are applied to sequences \((x\theta^ n)\). For some special \(\theta\) it is proved that this sequence is uniformly distributed modulo 1 if it is asymptotically distributed mod 1 to a measure the Fourier coefficients of which tend to 0.