2017/10/24 by Yeşim Demiroğlu Karabulut, Karabulut, Yeşim Demiroğlu
Mathematics · #05C50 (Primary) 16U60 #15B33 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA) #Spectral Theory (math.SP) #math.CO #math.RA #math.SP #msc:05C50 #msc:15B33 #msc:16U60
paper · pdf · doi:10.48550/arxiv.1710.08872
19 pages
arxiv created 2017/10/24 · arxiv updated 2017/10/25
We use the unit-graphs and the special unit-digraphs on matrix rings to show that every n × n nonzero matrix over \Bbb Fq can be written as a sum of two SLn-matrices when n>1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if X is a subset of Mat2 (\Bbb Fq) with size |X| > (2 q3 √(q))/(q - 1), then X contains at least two distinct matrices whose difference has determinant α for any α∈ \Bbb Fq∗. Using this result we also prove a sum-product type result: if A,B,C,D ⊆ \Bbb Fq satisfy √[4]|A||B||C||D|= Ω(q0.75) as q → ∞, then (A - B)(C - D) equals all of \Bbb Fq. In particular, if A is a subset of \Bbb Fq with cardinality |A| > \frac3 2 q(3)/(4), then the subset (A - A) (A - A) equals all of \Bbb Fq. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.