Corners in dense subsets of P^d

TL;DR

This paper proves that dense subsets of the d-fold product of primes contain infinitely many geometric configurations called corners, using a hypergraph removal lemma and linear forms.

Contribution

It introduces a removal lemma for weighted uniform hypergraphs based on linear forms, enabling the proof of corners in dense prime product sets.

Findings

Dense subsets of P^d contain infinitely many corners.

The method uses a hypergraph removal lemma with weights from linear forms.

Establishes a new approach linking hypergraph theory and number theory.

Abstract

Let $\PP^d$ be the $d$-fold direct product of the set of primes. We prove that if $A$ is a subset of $\PP^d$ of positive relative upper density then $A$ contains infinitely many "corners", that is sets of the form $\{x,x+te_1,...,x+te_d\}$ where x is an integer point and e_1,...,e_d are the standard basis vectors of the d-dimensional Euclidean space. Our argument is based on proving a removal lemma for weighted uniform hypergraphs, where the weight system is defined in terms of a pairwise linearly independent family of linear forms.