Euclidean sets with only one distance modulo a prime ideal

Abstract

Let X be a finite set in the Euclidean space $R^d$. If the squared distance between any two distinct points in X is an odd integer, then the cardinality of X is bounded above by $d+2$, as shown by Rosenfeld (1997) or Smith (1995). They proved that there exists a $(d+2)$-point set X in $R^d$ having only odd integral squared distances if and only if $d+2$ is congruent to 0 modulo 4. The distances can be interpreted as an element of the finite field $Z/2Z$. We generalize this result for a local ring $(A_p,p{A}_p)$ as follows. Let K be an algebraic number field that can be embedded into $R$. Fix an embedding of K into $R$, and K is interpreted as a subfield of $R$. Let $A=O_K$ be the ring of integers of K, and $p$ a prime ideal of $O_K$. Let $(A_p,p{A}_p)$ be the local ring obtained from the localization ${(A\setminus p)}^{-1}A$, which is interpreted as a subring of $R$. If the squared distances of $X\subset R^d$ are in $A_p$ and each squared distance is congruent to some constant $k\not\equiv 0$ modulo $p{A}_p$, then $\left|X\right|\le d+2$, as shown by Nozaki (2023). In this paper, we prove that there exists a set $X\subset R^d$ attaining the upper bound $\left|X\right|\le d+2$ if and only if $d+2$ is congruent to 0 modulo 4 when the finite field $A_p/p{A}_p$ is of characteristic 2, and $d+2$ is congruent to 0 modulo p when $A_p/p{A}_p$ is of characteristic p odd. We also provide examples attaining this upper bound.