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

We confirm a conjecture posed by Bergelson, Moreira, and Richter (arXiv:1711.05729), and in particular show that for every probability measure preserving system $(X,\mathcal{B},\mu ,T)$, every $k\in \mathbb{N}$, every set $A\in \mathcal{B}$ with $\mu (A)>0$, and every tempered function $f$, $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{n=1}^N\mu (A\cap T^{-\lfloor f(n)\rfloor }A\cap T^{-\lfloor f(n+1)\rfloor }A\cap \cdots \cap T^{-\lfloor f(n+k)\rfloor }A)>0.$$ This is achieved by establishing conditions on an increasing function $W:\mathbb{N}\to (0,\mathrm{\infty })$ such that if $(x_n{)}_{n\in \mathbb{N}}$ is a bounded sequence in a Banach space with $$\underset{W(N)-W(M)\to \mathrm{\infty }}{\lim }\frac{1}{W(N)-W(M)}\sum _{n=M}^N(W(n)-W(n-1))x_n=L$$ then the limit of Cesàro averages of $(x_n{)}_{n\in \mathbb{N}}$, ${\lim }_{N\to \mathrm{\infty }}\frac{1}{N}{\sum }_{n=1}^N{x}_n$ 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 $n\in [N-N^{1\mathrm{/}2+\epsilon },N]$ such that $$\{a,a+\lfloor n^{3\mathrm{/}2}\rfloor ,a+\lfloor (n+1{)}^{3\mathrm{/}2}\rfloor ,\dots ,a+\lfloor (n+k{)}^{3\mathrm{/}2}\rfloor \}\subseteq E\mathrm{.}$$