Let $A\subseteq\{1,...,N\}$ and $P_1,...,P_\ell\in\Z[n]$ with $P_i(0)=0$ and
$\deg P_i=k$ for every $1\leq i\leq\ell$.

We show, using Fourier analytic techniques, that for every $\VE>0$, there
necessarily exists $n\in\N$ such that \[\frac{|A\cap
(A+P_i(n))|}{N}>(\frac{|A|}{N})^2-\VE\] holds simultaneously for $1\leq i\leq
\ell$ (in other words all of the polynomial shifts of the set $A$ intersect $A$
"$\VE$-optimally"), as long as $N\geq N_1(\VE,P_1,...,P_\ell)$. The
quantitative bounds obtained for $N_1$ are explicit but poor; we establish that
$N_1$ may be taken to be a constant (depending only on $P_1,...,P_\ell$) times
a tower of 2's of height $C_{k,\ell}^*+C\eps^{-2}$.