A Multidimensional Szemer\'edi Theorem in the primes

TL;DR

This paper extends Green and Tao's theorem to higher dimensions, showing that dense subsets of prime tuples contain affine copies of any finite set, using a novel hypergraph approach without a transference principle.

Contribution

It introduces a new hypergraph method with non-uniform weights to prove a multidimensional Szemerédi theorem in the primes, extending previous results.

Findings

Dense prime subsets contain affine copies of any finite set in d6.

The hypergraph approach handles non-uniform weights on lower-dimensional edges.

The method avoids the transference principle, relying on the linear forms condition.

Abstract

Let $A$ be a subset of positive relative upper density of $\PP^d$, the $d$-tuples of primes. We prove that $A$ contains an affine copy of any finite set $F\subs\Z^d$, which provides a natural multi-dimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes. The proof uses the hypergraph approach by assigning a pseudo-random weight system to the pattern $F$ on a $d+1$-partite hypergraph; a novel feature being that the hypergraph is no longer uniform with weights attached to lower dimensional edges. Then, instead of using a transference principle, we proceed by extending the proof of the so-called hypergraph removal lemma to our settings, relying only on the linear forms condition of Green and Tao.