Applications of Cayley Digraphs to Waring’s Problem and Sum-Product Formulas (Yesim Demiroglu, Harvey Mudd)

Abstract: In this talk, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring’s problem over finite fields, and explain how in the process of re-proving these results, we obtain an original result that provides an analogue of Sarkozy’s theorem in the finite field setting (showing that any subset E of a finite field Fq for which |E| >  (qk)/sqrt{q – 1}must contain at least two distinct elements whose difference is a kth power). Once we have our results for finite fields, we apply some classical mathematics to extend our Waring’s problem results to the context of general (not  necessarily commutative) finite rings. In the second half of our talk, we present our sum-product results related to matrix rings over finite fields, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.