A maximal extension of the Bloom-Maynard bound for sets with no square differences

TL;DR

This paper extends bounds on the size of sets lacking differences of polynomial form, improving previous results and generalizing recent work by Bloom and Maynard.

Contribution

It provides a maximal extension of the Bloom-Maynard bound for sets with no polynomial difference, applicable to a broad class of polynomials.

Findings

Sets with no polynomial difference have density at most $(rac{1}{ ext{log} N})^{c ext{log} ext{log} ext{log} N}$

Improves the best known bounds in the literature

Generalizes recent results of Bloom and Maynard

Abstract

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N)^{-c\log\log\log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent result of Bloom and Maynard.