Weighted averages and applications to sets of multiple recurrence

We introduce new techniques for determining combinatorial properties of sets of multiple recurrence by considering weighted averages with quickly growing weights. Our main result is a far-reaching generalization of Szemerédi's Theorem which additionally confirms a conjecture of Bergelson-Moreira-Richter and contains as special cases both the Polynomial Szemerédi Theorem due to Bergelson-Leibman-Lesigne and the fact that if $f$ belongs to a broad class of smooth functions and satisfies $x^{d-1}\prec f(x)\prec x^d$ for some $d\in \mathbb{N}$ then for any $\ell\in \mathbb{N}$, any invertible measure preserving system $(X,\mathscr{B},μ,T)$, and any $A\in \mathscr{B}$ with $μ(A)>0$, the set $\{n\in \mathbb{N}: μ(A\cap T^{-[f(n)]}A\cap T^{-2[f(n)]}A\cap \cdots\cap T^{-\ell[f(n)]}A )>0\}$ is thick, meaning that it contains arbitrarily long intervals of natural numbers. Additionally, we formulate and prove a generalization to weighted averages of Boshernitzan's criterion for uniform distribution which we use in the proof of our main result.