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AbstractWe use the unit-graphs and the special unit-digraphs on matrix rings to show that every n×n{n\times n}nonzero matrix over 𝔽q{{\mathbb{F}}_{q}}can be written as a sum of two SLn{\operatorname{SL}_{n}}-matrices when n>1{n>1}. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and we prove that if X is a subset of Mat2(𝔽q){\operatorname{Mat}_{2}({\mathbb{F}}_{q})}with size |X|>2q3q/(q-1){\lvert X\rvert>2q^{3}\sqrt{q}/(q-1)}, then X contains at least two distinct matrices whose difference has determinant α for any α∈𝔽q∗{\alpha\in{\mathbb{F}}_{q}^{\ast}}. Using this result, we also prove a sum-product type result: if A,B,C,D⊆𝔽q{A,B,C,D\subseteq{\mathbb{F}}_{q}}satisfy |A||B||C||D|4=Ω(q0.75){\sqrt[4]{\lvert A\rvert\lvert B\rvert\lvert C\rvert\lvert D\rvert}=\Omega(q^{%0.75})}as q→∞{q\rightarrow\infty}, then (A-B)(C-D){(A-B)(C-D)}equals all of 𝔽q∗{{\mathbb{F}}_{q}^{\ast}}. In particular, if A is a subset of 𝔽q{{\mathbb{F}}_{q}}with cardinality |A|>32q3/4{\lvert A\rvert>\frac{3}{2}q^{3/4}}, then the subset (A-A)(A-A){(A-A)(A-A)}equals all of 𝔽q{{\mathbb{F}}_{q}}. We derive some identities involving character sums of the entries of 2×2{2\times 2}matrices over finite fields. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.
Forum Mathematicum – de Gruyter
Published: Nov 1, 2018
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