A new proof of Sarkozy's theorem

2011/07/01 by Neil Lyall, Lyall, Neil
Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CA #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1107.0243

to appear in the Proc. Amer. Math. Soc

arxiv created 2012/01/14 · arxiv updated 2012/01/17

Abstract

It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) 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.

Related

• Sarkozy's Theorem for P-Intersective Polynomials
• Polynomial Configurations in Difference Sets (Revised Version)
• Furstenberg--Sárközy theorem over number fields
• Sárközy's theorem in F₂[x]
• On complex positive definite functions on Z_n vanishing on squares