Let
$(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$
be an ergodic measure-preserving system, let
$A\in {\mathcal{B}}$
and let
$\unicode[STIX]{x1D716}>0$
. We study the largeness of sets of the form

$$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$
for various families

$\{f_{1},\ldots ,f_{k}\}$
of sequences

$f_{i}:\mathbb{N}\rightarrow \mathbb{N}$
. 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

$q(1)$
or

$q(-1)=0$
and

$p_{n}$
denotes the

$n$
th prime; or when

$f$
is a certain 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 abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the

$f_{i}$
are distinct polynomials.