Power savings for intersective polynomial differences and prime arguments

What OpenAI claims

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of {1,…,N} avoiding nonzero values h(1),h(2),… as differences has size O_h(N^1-c_k), with c_k > 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)

Papers (3)

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   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 > 0 such that every set A ⊆ {1,…,N} whose differences avoid all nonzero values h(1),h(2),…, 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.

2. A Power Saving for Polynomial Differences at Prime Arguments

   5 October 2026 · Preprint, not peer reviewed · PDF

   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 ⊆ {1,…,N} whose differences avoid all nonzero values h(p) at primes satisfies |A| ≤ C_hN^1-c_h, where c_h > 0 and C_h ≥ 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.

3. A power saving for square-difference-free sets

   24 September 2026 · Preprint, not peer reviewed · PDF

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

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.