On arithmetic progressions in nullspaces of integer matrices

TL;DR

This paper characterizes AP-sets as those with infinitely many solutions to certain linear systems, linking the existence of arithmetic progressions in nullspaces of integer matrices to the structure of AP-sets.

Contribution

It provides a new arithmetic characterization of AP-sets through solutions to homogeneous linear equations with zero column sum.

Findings

The vector (1,1,...,1) solves Mx=0 iff solutions with all coordinates in the same AP are infinite.

AP-sets are characterized by the existence of infinitely many solutions to specific linear systems.

The null space dimension being at least 2 is crucial for the characterization.

Abstract

Inspired by the Erd\"os-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem and S\'zmeredis Theorem that the primes and all subsets of positive upper density are AP-sets. We prove that (1, 1,..., 1) is a solution to the equation Mx = 0 where M is an integer matrix whose null space has dimension at least 2, if and only if the equation has infinitely many solutions such that the coordinates of each solution are elements in the same AP. This gives us a new arithmetic characterization of AP-sets, namely that they are the sets that have infinitely many solutions to a homogeneous system of linear equations, whenever the sum of the columns is zero.