  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.

Number Theory