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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 avoiding nonzero values as differences has size , with 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.
Manuscripts in this family · 3
A power saving for intersective polynomial differences with an exponent depending only on the degree
Oct 5, 2026·power-saving-intersective-polynomial-differences.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 such that every set whose differences avoid all nonzero values , where h is an intersective polynomial of degree k with positive leading coefficient, satisfies . 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
Oct 5, 2026·prime-argument-polynomial-differences.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 whose differences avoid all nonzero values at primes satisfies , where and 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
Sep 24, 2026·paper.pdf
We prove that there are absolute constants c > 0 and C < ∞ such that every set with no nonzero square difference satisfies . This answers the fixed-power question posed by Green and Sawhney.