A multi-dimensional Szemerédi theorem for the primes via a correspondence principle

2013/06/12 by Terence Tao, Tamar Ziegler, Tao, Terence +1
Mathematics · #11B30 #11T06 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11B30 #msc:11T06

paper · pdf · doi:10.48550/arxiv.1306.2886

20 pages, no figures. Submitted, Israel J. Math. Several suggestions of an anonymous referee have been implemented

openalex publication_date 2013/06/12 · arxiv created 2013/12/02 · arxiv updated 2013/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a version of the Furstenberg-Katznelson multi-dimensional Szemerédi in the primes \mathcal P := \2,3,5,…\, which roughly speaking asserts that any dense subset of \mathcal P^d contains constellations of any given shape. Our arguments are based on a weighted version of the Furstenberg correspondence principle, relative to a weight which obeys an infinite number of pseudorandomness (or "linear forms") conditions, combined with the main results of a series of papers by Green and the authors which establish such an infinite number of pseudorandomness conditions for a weight associated with the primes. The same result, by a rather different method, has been simultaneously established by Cook, Magyar, and Titichetrakun.

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