A new proof of Sárközy’s theorem

Lyall, Neil

doi:10.1090/S0002-9939-2013-11628-X

A new proof of Sárközy’s theorem
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Proc. Amer. Math. Soc. 141 (2013), 2253-2264 Request permission

Abstract:

It is a striking and elegant fact (proved independently by Furstenberg and Sárközy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth’s theorem.

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