Let
$P\in \mathbb{Z}\left[ n \right]$
with
$P(0)\,=\,0\,\text{and}\,\varepsilon \,>\,0$
. We show, using Fourier analytic techniques, that if
$N\ge \exp \exp \left( C{{\varepsilon }^{-1}}\log {{\varepsilon }^{-1}} \right)\,\text{and}\,A\,\subseteq \,\left\{ 1,\,.\,.\,.\,,\,N \right\}$
then there must exist
$n\in \mathbb{N}$
such that
$$\frac{\left| A\cap \left( A+P\left( n \right) \right) \right|}{N}>{{\left( \frac{\left| A \right|}{N} \right)}^{2}}-\,\varepsilon $$
.
In addition to this we show, using the same Fourier analytic methods, that if
$A\subseteq \mathbb{N}$
, then the set
of
$\varepsilon $
-optimal return times
$$R\left( A,P,\varepsilon \right)=\left\{ n\in \mathbb{N}:\delta \left( A\cap \left. \left( A+P\left( n \right) \right) \right)> \right.\delta {{\left( A \right)}^{2}}-\varepsilon \right\}$$
is syndetic for every
$\varepsilon >0$
. Moreover, we show that
$R\left( A,\,P,\,\varepsilon \right)$
is dense in every sufficiently long interval, in particular we show that there exists an
$L=L\left( \varepsilon ,P,A \right)$
such that
$$\left| R\left( A,P,\varepsilon \right)\cap I \right|\ge c\left( \varepsilon ,P \right)\left| I \right|$$
for all intervals
$I$
of natural numbers with
$\left| I \right|\,\ge \,L\,\text{and}\,c\left( \varepsilon ,\,P \right)\,=\,\exp \exp \,\left( -C\,{{\varepsilon }^{-1}}\,\log {{\varepsilon }^{-1}} \right).$