A multi-dimensional Szemer\'edi theorem for the primes via a correspondence principle

TL;DR

This paper proves a multi-dimensional Szemerédi theorem for primes, showing dense subsets contain all shapes, using a weighted correspondence principle and pseudorandomness conditions, extending combinatorial number theory into higher dimensions.

Contribution

It introduces a new multi-dimensional Szemerédi theorem for primes using a weighted correspondence principle and pseudorandomness conditions, building on prior work by Green and the authors.

Findings

Dense subsets of prime d-tuples contain all configurations of a given shape.

The proof employs a weighted version of the Furstenberg correspondence principle.

The result complements and extends prior work by Cook, Magyar, and Titichetrakun.

Abstract

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