Cayley Digraphs Associated to Arithmetic Groups

2018/08/20 by David Covert, Covert, David, Yeşim Demiroğlu Karabulut +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1808.06665

arxiv created 2018/08/20 · arxiv updated 2018/08/22

Abstract

We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called 2-simplices) in finite fields. Among other results we show that if \mathbbF_q is the finite field of odd order q, then every matrix in Mat_d(\mathbbF_q), d ≥ 2 is the sum of a certain (finite) number of orthogonal matrices, this number depending only on d, the size of the matrix, and on whether q is congruent to 1 or 3 (mod 4), but independent of q otherwise.

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