## Cayley digraphs of matrix rings over finite fields (Yesim Demiroglu, HMC)

### November 13 @ 12:15 pm - 1:10 pm

In this talk we use the unit-graphs and the special unit-digraphs on matrix rings to show that every n x n nonzero matrix over F_q can be written as a sum of two SL_n-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 Mat_2 (F_q) with size |X| > (2 q^3 \sqrt{q})/(q – 1), then X contains at least two distinct matrices whose difference has determinant $\alpha$ for any $\alpha \in F_q^*$. Using this result we also prove a sum-product type result: if $A,B,C,D \subseteq F_q$ satisfy $\sqrt[4]{|A||B||C||D|}= \Omega (q^{0.75})$ as q tends to infinity, then $(A – B)(C – D)$ equals all of $F_q$. In particular, if A is a subset of F_q with cardinality $|A| > \frac{3}{2} q^{3/4}$, then the subset $(A – A) (A – A)$ equals all of $F_q$. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units. This talk should be accessible to undergraduates with some background in linear algebra.