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.

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 estimates.

We prove that there are absolute constants c > 0 and C < ∞ such that every set $A\subseteq\{1,\ldots,N\}$ with no nonzero square difference satisfies $|A|\le C N^{1-c}$. This answers the fixed-power question posed by Green and Sawhney.