Binary Quadratic Forms in Difference Sets

Abstract

We show that if $h(x,y)=a{x}^2+bxy+c{y}^2\in \mathbb{Z}[x,y]$ satisfies $\Delta (h)=b^2-4ac\ne 0$, 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.

14 pages, typos corrected, to appear in Proceedings of Combinatorial and Additive Number Theory 2018