Dot-product sets and simplices over finite rings

Abstract

In this paper, we study dot-product sets and k-simplices in Z d n for odd n, where Z n is the ring of residues modulo n. We show that if E is sufficiently large then the dotproduct set of E covers the whole ring. In higher dimensional cases, if E is sufficiently large then the set of simplices and the set of dot-product simplices determined by E, up to congurence, have positive densities.