A Power Saving for Polynomial Differences at Prime Arguments

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 $A\subseteq \{1,\dots ,N\}$ whose differences avoid all nonzero values $h(p)$ at primes satisfies $|A|\le C_h{N}^{1-c_h}$, where $c_h>0$ and $C_h\ge 1$ 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.