Optimal lower bounds for multiple recurrence.

Let $(X,{\mathcal B},\mu,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $\epsilon>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \big\{n\in\mathbb{N}\colon\mu(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> \mu(A)^{k+1} - \epsilon \big\} \end{split} \end{equation*} for various families $(f_1,\dots,f_k)$ of functions $f_i:\mathbb{N}\to\mathbb{Z}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_n)$ where $q\in\mathbb{Z}[x]$ satisfies $q(1)=0$ and $(p_n)$ is the sequence of primes; or when $f$ is a Hardy field sequence. If $T^q$ is ergodic for some $q \in \mathbb{N}$, then for all $r \in \mathbb{Z}$, $S$ is syndetic if $f(n) = qn + r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and for $k = 3$ we found an interesting relation between the largeness of $S$ and the existence (and abundance) of solutions to certain linear equations in sparse sets of integers. We also provide several partial results when the $f_{i}$ are distinct polynomials.