Constellations in P^d

TL;DR

This paper proves that dense subsets of prime d-tuples contain affine copies of finite lattice point sets in general position, extending combinatorial number theory results to prime tuples.

Contribution

It establishes the existence of affine copies of finite lattice point sets within dense subsets of prime tuples in P^d, under a general position condition.

Findings

Dense subsets of P^d contain affine copies of finite lattice sets.

The result applies to sets with at most one point on each coordinate hyperplane.

It extends combinatorial number theory to prime tuples in higher dimensions.

Abstract

Let A be a subset of positive relative upper density of P^d, the d-tuples of primes. We prove that A contains an affine copy of any finite set of lattice points E, as long as E is in general position in the sense that it has at most one point on every coordinate hyperplane.