Single and multiple recurrence along non-polynomial sequences

Abstract

We establish new recurrence and multiple recurrence results for a rather large family $\mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $\ell \in \mathbb{N}\cup \{0\}$, ${\lim }_{x\to \infty }{f}^{(\ell )}(x)=\pm \infty$ and ${\lim }_{x\to \infty }{f}^{(\ell +1)}(x)=0$. Among other things, we show that for any $f\in \mathcal{F}$, any invertible probability measure preserving system $(X,\mathcal{B},\mu ,T)$, any $A\in \mathcal{B}$ with $\mu (A)>0$, and any $\epsilon >0$, the sets of returns $$R_{\epsilon ,A}=\{n\in \mathbb{N}:\mu (A\cap T^{-\lfloor f(n)\rfloor }A)>{\mu }^2(A)-\epsilon \}$$ and $$R_A^{(k)}=\{n\in \mathbb{N}:\mu (A\cap T^{\lfloor f(n)\rfloor }A\cap T^{\lfloor f(n+1)\rfloor }A\cap \cdots \cap T^{\lfloor f(n+k)\rfloor }A)>0\}$$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.

51 pages

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