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On solutions in arithmetic progressions to homogenous systems of linear equations

By Jonas Lindstrøm Jensen
No. 02, February 2009

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 that the primes is an AP-set. We prove that the equation

$ M\underline{x} = 0, $

where $M$ is an integer matrix whose null space has dimension at least $2$, has infinitely many solutions in any AP-set such that the coordinates of each solution are elements in the same AP, if and only if $(1,1,\ldots,1)$ is a solution.

We will furthermore prove that AP-sets are exactly the sets that has infinitely many solutions to a homogeneous system of linear equations, whenever the sum of the columns is zero.

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