Results on the Erdős–Falconer distance problem in [formula omitted] for odd q

Abstract

The Erdős–Falconer distance problem in Zqd asks one to show that if E⊂Zqd is of sufficiently large cardinality, then the set of distances determined by E satisfies Δ(E)=Zq. Previous results were known only in the case q=pℓ, where p is an odd prime, and as such only showed that all units were obtained in the distance set. We give the first such result over rings Zq where q is no longer confined to be a prime power, and despite this, we show that the distance set of E contains all of Zq whenever E is of sufficiently large cardinality.