Uniform Weighted Averages and a Conjecture of Bergelson, Moreira, and Richter

Abstract

We confirm a conjecture posed by Bergelson, Moreira, and Richter (arXiv:1711.05729), and in particular show that for every probability measure preserving system $(X,\mathscr{B},\mu,T)$, every $k\in \mathbb{N}$, every set $A\in \mathscr{B}$ with $\mu(A)>0$, and every tempered function $f$, \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu(A\cap T^{-\lfloor{f(n)\rfloor}}A\cap T^{-\lfloor{f(n+1)\rfloor}}A\cap \cdots \cap T^{-\lfloor{f(n+k)\rfloor}}A)>0. \] This is achieved by establishing conditions on an increasing function $W:\mathbb{N}\rightarrow (0,\infty)$ such that if $(x_n)_{n\in \mathbb{N}}$ is a bounded sequence in a Banach space with \[ \lim_{W(N)-W(M)\to\infty}\frac{1}{W(N)-W(M)}\sum_{n=M}^N (W(n)-W(n-1))x_n =L \] then the limit of Ces\`aro averages of $(x_n)_{n\in \mathbb{N}}$, $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nx_n$ is also equal to $L$. Furthermore, the methods we develop can be used to sharpen some of the combinatorial results obtained by Bergelson, Moreira, and Richter. For example, if $E$ is a set of positive upper density, then for any $k\in \mathbb{N}$, any $\epsilon>0$, and all sufficiently large $N\in \mathbb{N}$ there is an $n\in [N-N^{1/2+\epsilon},N]$ such that \[\{a,a+\lfloor{n^{3/2}\rfloor},a+\lfloor{(n+1)^{3/2}\rfloor},\dots ,a +\lfloor{(n+k)^{3/2}\rfloor}\}\subseteq E. \]