Power savings for intersective polynomial differences and prime arguments

An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent c_k> 0 such that every set A⊆\{1,\ldots,N\} whose differences avoid all nonzero values h(1),h(2),\ldots, where h is an intersective polynomial of degree k with positive leading coefficient, satisfies |A|=O_h(N^{1-c_k}). The implied constant may depend on h, but the power-saving exponent depends only on its degree.

Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set A⊆\{1,\ldots,N\} whose differences avoid all nonzero values h(p) at primes satisfies |A|≤ C_hN^{1-c_h}, where c_h> 0 and C_h\ge1 depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.