Binary Quadratic Forms in Difference Sets

Abstract

We show that if h(x, y) = ax 2 + bxy + cy 2 ∈ Z[x, y] satisfies ∆(h) = b 2 − 4ac = 0, then any subset of {1, 2, . . . , N } lacking nonzero differences in the image of h has size at most a constant depending on h times N exp(−c √ 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.