A power saving for intersective polynomial differences with an exponent depending only on the degree
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 $c_k\gt 0$ such that every set $A\subseteq\{1,\ldots,N\}$ 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.
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