Bounds for sets with few distances distinct modulo a prime ideal

Abstract

Let 𝒪 K be the ring of integers of an algebraic number field K embedded into ℂ. Let X be a subset of the Euclidean space ℝ d , and D(X) be the set of the squared distances of two distinct points in X. In this paper, we prove that if D(X)⊂𝒪 K and there exist s values a 1 ,...,a s ∈𝒪 K distinct modulo a prime ideal 𝔭 of 𝒪 K such that each a i is not zero modulo 𝔭 and each element of D(X) is congruent to some a i , then |X|≤d+s s+d+s-1 s-1.