We confirm a conjecture posed by Bergelson, Moreira, and Richter, 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 }({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 }({1}/({W(N)-W(M)})) \sum _{n=M}^N (W(n)-W(n-1))x_n =L $ then the limit of Cesàro averages of $(x_n)_{n\in \mathbb {N}}$, $\lim _{N\to \infty }({1}/{N})\sum _{n=1}^Nx_n$, 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 $\varepsilon>0$, and all sufficiently large $N\in \mathbb {N}$ there is an $a\in \mathbb {N}$ and an $n\in [N-N^{1/2+\varepsilon },N]$ such that $\{a,a+\lfloor n^{3/2} \rfloor ,a+\lfloor (n+1)^{3/2}\rfloor ,\ldots ,a +\lfloor (n+k)^{3/2}\rfloor \}\subseteq E.$