Abundance of Matrices In Gaussian Integers

Abstract

In [HLS], N. Hindman, I. Leader and D. Strauss proved the abundance for a matrix with rational entries. In this paper we proved it for the ring of Gaussian integers. We showed the result when the matrix is taken with entries from \mathbb{Q}\left[i\right]. The main obstacle is in the field of complex numbers, no linear order relation exists. We overcome that in a tactful way.