Sarkozy's Theorem for P-Intersective Polynomials
Abstract
We define a necessary and sufficient condition on a polynomial h∈ 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,…,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 ( N)-c for any c<1/(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.
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