Binary Quadratic Forms in Difference Sets

TL;DR

This paper proves that subsets of initial segments of natural numbers avoiding differences in the image of certain binary quadratic forms are very small, using an adapted density increment method.

Contribution

It introduces a new application of an $L^2$ density increment strategy to binary quadratic forms with nonzero discriminant, extending previous polynomial sum results.

Findings

Subsets avoiding differences in quadratic form images are exponentially small.

The method adapts polynomial sum techniques to quadratic forms.

Provides a thorough, accessible exposition of the adapted strategy.

Abstract

We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies $\Delta(h)=b^2-4ac\neq 0$, then any subset of $\{1,2,\dots,N\}$ lacking nonzero differences in the image of $h$ has size at most a constant depending on $h$ times $N\exp(-c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an $L^2$ density increment strategy previously used to establish analogous results for sums of one or more single-variable polynomials. Our exposition is thorough and self-contained, in order to serve as an accessible gateway for readers who are unfamiliar with previous implementations of these techniques.