A short proof of the multidimensional Szemer\'edi theorem in the primes

TL;DR

This paper provides a simplified proof of Tao's conjecture that dense subsets of prime $d$-tuples contain any given constellation shape, leveraging existing theorems in primes and multidimensional combinatorics.

Contribution

It offers a straightforward proof of the multidimensional Szemerédi theorem in primes, building on the Green-Tao and Furstenberg-Katznelson theorems.

Findings

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

The proof simplifies previous complex arguments.

It confirms Tao's conjecture using established theorems.

Abstract

Tao conjectured that every dense subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This was very recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler. Here we give a simple proof using the Green-Tao theorem on linear equations in primes and the Furstenberg-Katznelson multidimensional Szemer\'edi theorem.