IP-systems, generalized polynomials and recurrence

Abstract

Let $\Omega$ be an abelian group. A set $R\subset \Omega$ is a set of recurrence if, for any probability measure-preserving system $(X,{\mathcal B},\mu,\{T_g\}_{g\in \Omega})$ and any $A\in {\mathcal A}$ with $\mu(A)>0,\ \mu(A\cap T_g A)>0$ for some $g\in R$. If $(x_i)_{i=1}^\infty$ is a sequence in $\Omega$, the set of its finite sums $\{ x_{i_1}+x_{i_2}+\dotsb + x_{i_k}: i_1<i_2<\dotsb <i_k\}$ is called an IP-set. In Bergelson et al (Erg. Theor. Dyn. Sys.16 (1996), 963–974) it is shown that if $p:{\mathbb Z}^d{\rightarrow} {\mathbb Z}^k$ is a polynomial vanishing at zero and F is an IP-set in ${\mathbb Z}^d$ then $\{ p(n):n\in F\}$ is a set of recurrence in ${\mathbb Z}^k$. Here we extend this result to an analogous family of generalized polynomials, that is functions formed from regular polynomials by iterated use of the greatest integer function, as a consequence of a theorem establishing a much wider class of recurrence sets occurring in any (possibly non-finitely generated) abelian group. While these sets do in a sense have a distinctively ‘polynomial’ nature, this far-ranging class includes, even in ${\mathbb Z}$, such examples as $\big\{\!\sum_{i, j\in \alpha, i<j} 2^i3^j : \alpha\subset \mathbb{N}, 0<|\alpha|<\infty\big\}$, where the connection to conventional polynomials is somewhat distant.