Binary Quadratic Forms in Difference Sets

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