Power savings for intersective polynomial differences and prime arguments

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of \{1,\ldots,N\} avoiding nonzero values h(1),h(2),\ldots as differences has size O_h(N^{1-c_k}), with c_k\gt 0 depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h. (Lean)

Manuscripts (3)

A power saving for intersective polynomial differences with an exponent depending only on the degree

October 5, 2026 · source · cite as \cite{OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026}

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.

A Power Saving for Polynomial Differences at Prime Arguments

October 5, 2026 · source · cite as \cite{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026}

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.

A power saving for square-difference-free sets

September 24, 2026 · source · cite as \cite{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026}

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.

Result 182 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.