Bounds for sets with few distances distinct modulo a prime ideal

TL;DR

This paper establishes upper bounds on the size of point sets in Euclidean space with squared distances constrained to a subset of algebraic integers, specifically when distances are limited modulo a prime ideal.

Contribution

It provides new bounds for the size of sets with distances in algebraic integers constrained modulo a prime ideal, extending combinatorial geometry to algebraic number fields.

Findings

Sets with distances in $\\mathcal{O}_K$ are bounded in size by combinatorial expressions.

The bounds depend on the dimension and the number of distinct residue classes modulo a prime ideal.

The results generalize classical distance set problems to algebraic number fields.

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 $|X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}$.