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 $A\subseteq\{1,\ldots,N\}$ whose differences avoid all nonzero values $h(p)$ at primes satisfies $|A|\le C_hN^{1-c_h}$, where $c_h\gt 0$ 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…
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