Abstract
We show that if \(h(x,y)=ax^2+bxy+cy^2\in \mathbb {Z}[x,y]\) satisfies \(\varDelta (h)=b^2-4ac\ne 0,\) then any subset of \(\{1,2,\ldots ,N\}\) 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.
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Acknowledgements
The author would like to thank Neil Lyall who co-authored the expository note [12], in the context of squares and shifted primes, that served as a template for this paper.
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Rice, A. (2020). Binary Quadratic Forms in Difference Sets. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory III. CANT 2018. Springer Proceedings in Mathematics & Statistics, vol 297. Springer, Cham. https://doi.org/10.1007/978-3-030-31106-3_14
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DOI: https://doi.org/10.1007/978-3-030-31106-3_14
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