Sarkozy's Theorem for P-Intersective Polynomials

We define a necessary and sufficient condition on a polynomial $h\in $\mathbb{Z}$[x]$ to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form $h(p)$ for some prime $p$. Moreover, we establish a quantitative estimate on the size of the largest subset of $1,2,\dots ,N$ which lacks the desired arithmetic structure, showing that if deg$(h)=k$, then the density of such a set is at most a constant times $(\log N{)}^{-c}$ for any $c<1\mathrm{/}(2k-2)$. We also discuss how an improved version of this result for $k=2$ and a relative version in the primes can be obtained with some additional known methods.