A new proof of Roth's theorem on arithmetic progressions

2008/01/16 by Ernie Croot, Croot, Ernie, Olof Sisask +1
Mathematics · #05D99 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05D99

paper · pdf · doi:10.48550/arxiv.0801.2577

6 pages. To appear in Proceedings of the AMS

arxiv created 2008/04/01 · arxiv updated 2009/12/01

Abstract

We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs of Roth's theorem), we do not pass to a progression related to the large Fourier coefficients of our set (as most other proofs of Roth do). Furthermore, in our proof, the density increment is achieved through an application of a quantitative version of Varnavides's theorem, which is perhaps unexpected.

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