Euclidean sets with only one distance modulo a prime ideal

Let $X$ be a finite set in the Euclidean space $\mathbb{R}^d$. 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 $\mathbb{R}^d$ 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 $\mathbb{Z}/2\mathbb{Z}$. We generalize this result for a local ring $(A_\mathfrak{p},\mathfrak{p}A_\mathfrak{p})$ as follows. Let $K$ be an algebraic number field that can be embedded into $\mathbb{R}$. Fix an embedding of $K$ into $\mathbb{R}$, and $K$ is interpreted as a subfield of $\mathbb{R}$. Let $A=O_K$ be the ring of integers of $K$, and $\mathfrak{p}$ a prime ideal of $O_K$. Let $(A_\mathfrak{p},\mathfrak{p}A_\mathfrak{p})$ be the local ring obtained from the localization $(A\setminus \mathfrak{p})^{-1} A$, which is interpreted as a subring of $\mathbb{R}$. If the squared distances of $X\subset \mathbb{R}^d$ are in $A_\mathfrak{p}$ and each squared distance is congruent to some constant $k \not\equiv 0$ modulo $\mathfrak{p} A_\mathfrak{p}$, then $|X| \leq d+2$, as shown by Nozaki (2023). In this paper, we prove that there exists a set $X\subset \mathbb{R}^d$ attaining the upper bound $|X| \leq d+2$ if and only if $d+2$ is congruent to $0$ modulo $4$ when the finite field $A_\mathfrak{p}/ \mathfrak{p} A_\mathfrak{p}$ is of characteristic 2, and $d+2$ is congruent to $0$ modulo $p$ when $A_\mathfrak{p}/ \mathfrak{p} A_\mathfrak{p}$ is of characteristic $p$ odd. We also provide examples attaining this upper bound.