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 $a_1,\ldots, a_s \in \mathcal{O}_K$ 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 $|X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}$.