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

Abstract

We establish a version of the Furstenberg-Katznelson multi-dimensional Szemerédi in the primes $\mathcal{P}:=\{2,3,5,\dots \}$, 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.

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

References in corpus (1)

• A relative Szemerédi theorem

Cited by in corpus (1)

• The Green-Tao theorem: an exposition