Bounds for sets with few distances distinct modulo a prime ideal

Abstract

Let ${\mathcal{O}}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space ${\mathbb{R}}^d$, and $D(X)$ be the set of the squared distances of two distinct points in $X$. In this paper, we prove that if $D(X)\subset {\mathcal{O}}_K$ and there exist $s$ values $a_1,\dots ,a_s\in {\mathcal{O}}_K$ distinct modulo a prime ideal $\mathfrak{p}$ of ${\mathcal{O}}_K$ such that each $a_i$ is not zero modulo $\mathfrak{p}$ and each element of $D(X)$ is congruent to some $a_i$, then $|X|\le \left(\genfrac{}{}{0ex}{}{d+s}{s}\right)+\left(\genfrac{}{}{0ex}{}{d+s-1}{s-1}\right)$.

arXiv comment8 pages, no figure