Results on the Erdős–Falconer distance problem in Zqd for odd q

Abstract

The Erdős-Falconer distance problem in Z d q asks one to show that if E ⊂ Z d q is of sufficiently large cardinality, then ∆(E) :Here, Z q is the set of integers modulo q, and Z d q = Z q × · · · × Z q is the free module of rank d over Z q . We extend known results in two directions. Previous results were known only in the setting q = p ℓ , where p is an odd prime, and as such only showed that all units were obtained in the distance set. We remove the constriction that q is a power of a prime, and despite this, shows that the distance set of E contains all of Z q whenever E is of sufficiently large cardinality.

Dot-product sets and simplices over finite rings

“…In [17], the second listed author also studied this result over the ring of residues modulo n for an arbitrary n. In this paper, we will further extend the problem to cover the whole ring. Note that, our result is in line with the result of Covert in [5] for the Erdős distinct distances problem.…”

“…then the distance set determined by E will cover all units in Z p l . In [5], Covert extended the problem the ring of residues modulo n for an arbitrary odd n. Let p be the smallest prime divisor of n and τ (n) be the number of divisors of n, Covert showed that if…”

On the Sums of Any $k$ Points in Finite Fields

“…For further discussion on distance problems in finite fields, readers may refer to [5,9,16,17,18,19,26,27]. See also [3,4], and references contained therein for recent results on the distance problems in the ring setting.…”