\documentclass[ALCO, Unicode]{cedram} \usepackage[mathscr]{eucal} \usepackage{amssymb} \usepackage{latexsym} \usepackage{amsthm} \usepackage{amsmath} %\usepackage[dvips]{graphicx} \usepackage{psfrag} \def\labelenumi {$(\theenumi)$} \def\theenumii {\arabic{enumii}} \def\labelenumii {(\theenumii)} \newcommand{\mb}{\mathbf} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\A}{\boldsymbol{A}} \newcommand{\B}{\boldsymbol{B}} \newcommand{\E}{\boldsymbol{E}} \newcommand{\J}{\boldsymbol{J}} \newcommand{\I}{\boldsymbol{I}} \newcommand{\D}{\boldsymbol{D}} \newcommand{\F}{\boldsymbol{F}} \newcommand{\M}{\boldsymbol{M}} \newcommand{\Pm}{\boldsymbol{P}} \newcommand{\Sm}{\boldsymbol{S}} \newcommand{\X}{\boldsymbol{X}} \newcommand{\x}{\boldsymbol{x}} \newcommand{\y}{\boldsymbol{y}} \newcommand{\av}{\mathfrak{a}} \newcommand{\vf}{\mathfrak{v}} \newcommand{\jv}{\boldsymbol{j}} \newcommand{\vv}{\boldsymbol{v}} \newcommand{\uv}{\boldsymbol{u}} \newcommand{\wv}{\boldsymbol{w}} \newcommand{\el}{\boldsymbol{\ell}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \equalenv{remark}{rema} \equalenv{example}{exam} \equalenv{corollary}{coro} \makeatletter \@addtoreset{equation}{section} \def\theequation{\thesection.\arabic{equation}} \makeatother \title [Bounds for sets with few distances distinct modulo a prime ideal] {Bounds for sets with few distances distinct modulo a prime ideal} \author[\initial{H.} Nozaki]{\firstname{Hiroshi} \lastname{Nozaki}} \address{Aichi University of Education\\ Department of Mathematics Education\\ 1 Hirosawa, Igaya-cho\\ Kariya, Aichi\\ 448-8542 (Japan)} \email{hnozaki@auecc.aichi-edu.ac.jp} \keywords{$s$-distance set, algebraic number field} \subjclass{05D05, 05B30} \begin{document} \begin{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 $|X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}$. \end{abstract} \maketitle \section{Introduction} This paper is devoted to giving an upper bound on the cardinalities of certain finite sets $X$ in a metric space $M$, that have some special properties of the values of distances appearing in $X$. A finite set $X$ in $M$ is called an {\it $s$-distance set} if the number of distances of two distinct points in $X$ is equal to $s$. One of the major problems for $s$-distance sets is to determine the largest possible $s$-distance sets for given $s$, that is motivated from the extremal set theory. For this purpose, we need to give (or improve) upper bounds on the size and construct large sets. For small $s$, there are remarkable successful cases that can determine the largest sets, for example, $2$-distance sets on a Euclidean sphere \cite{GY18}, sets of equiangular lines in Euclidean spaces \cite{JTYZZ21}, and several $s$-distance sets in the real, complex, or quaternionic projective spaces \cite{L92} or in polynomial association schemes \cite{D73,DGS77,DL98}. In the literature of combinatorial geometry, upper bounds for $s$-distance sets for $L$-intersecting families have been obtained. An {\it $L$-intersecting family} $\mathfrak{F}$ is a family of subsets of a finite set $F$ that satisfies $|A \cap B| \in L$ for any distinct $A,B \in \mathfrak{F}$ for some $L\subset \{0,1,\ldots, n-1\}$, where $|F|=n$. An $L$-intersecting family $\mathfrak{F}$ is said to be {\it $k$-uniform} if $|A|=k$ for each $A \in \mathfrak{F}$ for some constant $k$. For $k$-uniform $L$-intersecting families $\mathfrak{F}$, Ray-Chaudhuri and Wilson \cite{R-CW75} proved an upper bound $|\mathfrak{F}| \leq \binom{n}{s}$, where $|L|=s$. This case corresponds to $s$-distance sets in Johnson association schemes. After this work, Frankl and Wilson \cite{FW81} obtained $|\mathfrak{F}|\leq \sum_{i=0}^s\binom{n}{i}$ without the assumption of $k$-uniform. Frankl and Wilson \cite{FW81} also proved a {\it modular version} of the upper bound for $k$-uniform $L$-intersecting families. Namely they interpreted the sizes of the intersections as elements of $\mathbb{Z}/p \mathbb{Z}$ for some prime number $p$. Suppose the set $L$ has only $s$ elements distinct modulo $p$, and note that $|L|$ may be greater than $s$. For $k$-uniform $L$-intersecting families $\mathfrak{F}$, if $k \not \equiv a \pmod{p}$ for each $a \in L$, then Frankl--Wilson \cite{FW81} showed that $|\mathfrak{F}| \leq \binom{n}{s}$. Note that this is the same upper bound as that obtained under the assumption $|L|=s$. For $L$-intersecting families $\mathfrak{F}$ with $r$ different sizes of elements of $\mathfrak{F}$ modulo $p$, Alon, Babai, and Suzuki \cite{ABS91} proved that $|\mathfrak{F}|\leq \sum_{i=0}^{r-1}\binom{n}{s-i}$ under a certain weak assumption which is simplified by \cite{HK15}. The upper bound in \cite{ABS91} is proved by Koornwinder's method \cite{K77}, which gives upper bounds on the size $|X|$ by proving the linear independence of some polynomial functions that have a bijective correspondence to $X$. We have upper bounds for Euclidean $s$-distance sets with several conditions, which are counterparts of that of $L$-intersecting families. Let $X$ be an $s$-distance set in the Euclidean space $\mathbb{R}^d$. For $X$ in the unit sphere $S^{d-1}$, which corresponds to the condition of $k$-uniform, Delsarte, Goethals, and Seidel \cite{DGS77} proved that $|X| \leq \binom{d+s-1}{s}+\binom{d+s-2}{s-1}$. With no assumption, Bannai, Bannai, and Stanton \cite{BBS83} proved that $|X| \leq \binom{d+s}{s}$. For $X$ in $r$ concentric spheres, which corresponds to the condition of $r$ different sizes, Bannai, Kawasaki, Nitamizu, and Sato \cite{BKNS03} obtained that $|X| \leq \sum_{i=0}^{2r-1}\binom{d+s-1-i}{s-i}$. Recently simple alternative proofs of these upper bounds are given in \cite{HR21,PP21}. Blokhuis \cite{BT} gave a modular version of upper bounds for Euclidean sets, assuming that the squared distances are rational integers. Let $D(X)$ be the set of the squared Euclidean distances between two distinct points of $X$. \begin{theorem}[{mod-$p$ bound \cite{BT}}] \label{thm:modp} Let $X$ be a subset of $\mathbb{R}^d$, and $p$ a prime number. Suppose $D(X)$ is a subset of rational integers $\mathbb{Z}$. If there exist $a_1,\ldots, a_s \in \mathbb{Z}$ distinct modulo $p$ such that \begin{enumerate} \item for each $i \in \{1,\ldots,s\}$, $a_i \not \equiv 0 \pmod{p}$ and \item for each $\alpha \in D(X)$, there exists $i \in \{1,\ldots,s\}$ such that $\alpha \equiv a_i \pmod{p}$, \end{enumerate} then \[ |X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}. \] \end{theorem} For sets in a sphere, several projective spaces, or $Q$-polynomial association schemes, Theorem~\ref{thm:modp} can be analogously obtained. However, the $r$-concentric spherical version is still open. The assumption $D(X) \subset \mathbb{Z}$ in Theorem~\ref{thm:modp} is surely strong, and the sets to which the theorem can be applied are restricted. In this paper, we extend Theorem~\ref{thm:modp} to the ring of integers $\mathcal{O}_K$ of an algebraic number field $K$, and any prime ideal $\mathfrak{p}$ of it. Note that throughout this paper, we fix an embedding of $K$ into $\mathbb{C}$, and $K$ is interpreted as a subfield of $\mathbb{C}$. We use Koornwinder's method to prove this mod-$\mathfrak{p}$ upper bound. However the method to prove the linear independence of polynomial functions is new. In the proof, the localization $A_{\mathfrak{p}}$ of $A=\mathcal{O}_K$ by a prime ideal $\mathfrak{p}$ plays a key role and Nakayama's lemma is applied for a certain finitely generated $A_{\mathfrak{p}}$-module. This method is purely algebraic and uniformly applicable to the polynomial spaces \cite{G89}, \cite[Sections 14--16]{Gb}, which includes the Euclidean sphere, the real, complex or quaternionic projective spaces (see \cite{DL98,L92} for the theory of $s$-distance set in these projective spaces), or $Q$-polynomial association schemes (which include the theory of $L$-intersecting family as codes of Johnson or Hamming schemes) \cite{D73}. The paper is organized as follows. In Section \ref{sec:2}, we introduce basic terminology and results about algebraic number fields. In Section \ref{sec:3}, we prove a generalization of Theorem~\ref{thm:modp} (mod-$\mathfrak{p}$ bound) for the ring of integers $\mathcal{O}_K$ of an algebraic number field $K$ and a prime ideal $\mathfrak{p}\subset \mathcal{O}_K$. We also comment on the version of the theorem for an ideal that may not be prime. In Section~\ref{sec:4}, we extend the LRS type theorem proved in \cite{LRS77,N11}. Namely, if the cardinality of an $s$-distance set is relatively large, then a certain ratio of squared distances must be an algebraic integer (see Theorem~\ref{thm:LRS}). We explain the relationship between the LRS type theorem and mod-$\mathfrak{p}$ bound, which can refine an upper bound on the size of an $s$-distance set with given distances. \section{Preliminaries} \label{sec:2} %In this section, we give basic terminologies and its properties that are used in the paper. An extension field $K$ of rationals $\mathbb{Q}$ is an {\it algebraic number field} if the degree $[K:\mathbb{Q}]$ is finite. An algebraic number field $K$ can be embedded into $\mathbb{C}$, and $K$ is always identified with a fixed specific subfield of $\mathbb{C}$. The {\it ring of integers $\mathcal{O}_K$} is the ring consisting of all algebraic integers in $K$, where an {\it algebraic integer} is a complex number which is a root of a monic polynomial with integer coefficients. It is well known that $K$ is the quotient field of $\mathcal{O}_K$, a prime ideal of $\mathcal{O}_K$ is maximal, $\mathcal{O}_K$ is a finitely generated free $\mathbb{Z}$-module, and $\mathcal{O}_K$ may not be a principal ideal domain. For easy examples, if $K=\mathbb{Q}$, then $\mathcal{O}_K=\mathbb{Z}$. If $K=\mathbb{Q}(\sqrt{d})$ for a square-free integer $d$, then \[ \mathcal{O}_K=\left\lbrace \begin{array}{ll} \mathbb{Z}+ \frac{1+\sqrt{d}}{2} \mathbb{Z} & \text{ if $ d \equiv 1 \!\pmod{4}$},\\ \mathbb{Z}+ \sqrt{d} \mathbb{Z} & \text{ if $ d \equiv 2,3 \!\pmod{4}$}. \end{array}\right. \] Suppose a ring is commutative and contains the identity. For a ring $A$, $(A,\mathfrak{m})$ is a {\it local ring} if $A$ has a unique maximal ideal $\mathfrak{m}$. It is well known that for a ring $A$ and its maximal ideal $\mathfrak{p}\subset A$, we can construct a local ring $(A_{\mathfrak{p}}, \mathfrak{p} A_{\mathfrak{p}})$. For $A=\mathcal{O}_K$, the local ring is \[ A_\mathfrak{p}=S^{-1} A=\{a/s \in K \mid a \in A, s \in S\}, \] where $S=A \setminus \mathfrak{p}$. Its unique maximal ideal is $\mathfrak{p} A_{\mathfrak{p}}$, which is the ideal of $A_{\mathfrak{p}}$ generated by the elements of $\mathfrak{p}$. Note that $A_\mathfrak{p}$ is a principal ideal domain. The natural map $f: A/\mathfrak{p} \rightarrow A_\mathfrak{p}/ \mathfrak{p} A_\mathfrak{p}$ is a field isomorphism. The following theorem is called Nakayama's lemma, which plays a key role in a proof of the main theorem. For a local ring $(A,\mathfrak{m})$, the ideal $I$ in Theorem~\ref{thm:nakayama} is $\mathfrak{m}$. \begin{theorem}\label{thm:nakayama} Let $A$ be a ring. Let $I$ be an ideal that is contained in all maximal ideals of $A$. Let $M$ be a finitely generated $A$-module. If $IM=M$, then $M=\{0\}$. \end{theorem} In order to prove the main theorem, we use the polynomial \[ f_{\x}(\boldsymbol{\xi})=\prod_{i=1}^s (||\x-\boldsymbol{\xi}||^2-a_i), \] for $\x \in \mathbb{R}^d$, $a_i \in \mathbb{R}$, and variables $\boldsymbol{\xi}=(\xi_1,\ldots, \xi_d)$, where $||\x||$ is the Euclidean norm of $\x$. By proving the linear independence of $\{f_{\x}\}_{\x \in X}$ as polynomial functions, the cardinality $|X|$ can be bounded above by the dimension of a certain linear space that contains $f_{\x}$. We use the same polynomial space used in Bannai--Bannai--Stanton \cite{BBS83}. For $\xi_0=\xi_1^2+\cdots + \xi_d^2$, we define the polynomial space $P_s(\mathbb{R}^d)$ that consists of all real polynomial functions on $\mathbb{R}^d$ which are spanned by $\xi_0^{\lambda_0}\xi_1^{\lambda_1}\cdots \xi_d^{\lambda_d}$ with $\sum_{i=0}^d\lambda_i\leq s$. The dimension of $P_s(\mathbb{R}^d)$ is equal to $\binom{d+s}{s}+\binom{d+s-1}{s-1}$. \section{Bounds for $s$-distance sets modulo $\mathfrak{p}$} \label{sec:3} The following is the main theorem in this paper. \begin{theorem}[mod-$\mathfrak{p}$ bound] \label{thm:main} Let $X$ be a subset of $\mathbb{R}^d$, and $A=\mathcal{O}_K$ the ring of integers of an algebraic number field $K$. Let $\mathfrak{p}$ be a prime ideal of $A$. Suppose $D(X) \subset A_{\mathfrak{p}}$. If there exist $a_1,\ldots, a_s \in A_\mathfrak{p}$ distinct modulo $\mathfrak{p}A_{\mathfrak{p}}$ such that \begin{enumerate} \item for each $i \in \{1,\ldots,s\}$, $a_i \not \equiv 0 \pmod{\mathfrak{p}A_\mathfrak{p}}$ and \item for each $\alpha \in D(X)$, there exists $i \in \{1,\ldots,s\}$ such that $\alpha \equiv a_i \pmod{\mathfrak{p}A_{\mathfrak{p}}}$, \end{enumerate} then \[ |X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}. \] \end{theorem} \begin{proof} For each $\x \in X$, we define the polynomial $f_{\x}(\boldsymbol{\xi})\in P_s(\mathbb{R}^d)$ as \[ f_{\x}(\boldsymbol{\xi})=\prod_{i=1}^s (||\x-\boldsymbol{\xi}||^2-a_i), \] where if needed, we replace $a_i$ with a real value equivalent to $a_i$ modulo $\mathfrak{p}A_{\mathfrak{p}}$. These polynomials satisfy \begin{equation} \label{eq:f_x(x)} f_{\x}(\boldsymbol{x})=(-1)^s \prod_{i=1}^s a_i \not\equiv 0 \pmod{\mathfrak{p}A_{\mathfrak{p}}}, \end{equation} and \begin{equation} \label{eq:f_x(y)} f_{\x}(\boldsymbol{y}) \equiv 0 \pmod{\mathfrak{p}A_{\mathfrak{p}}} \end{equation} for $\x \ne \y \in X$. We prove $\{f_{\x}\}_{\x \in X}$ is linearly independent as polynomial functions on $\mathbb{R}^d$. Assume there exist $m_{\x} \in \mathbb{R}$ such that \begin{equation} \label{eq:=0} \sum_{\x \in X} m_{\x} f_{\x}(\boldsymbol{\xi})=0. \end{equation} Let $M$ be an $A_\mathfrak{p}$-module generated by a finite set $\{m_{\x} \}_{\x \in X}$, namely \[ M=\sum_{\x \in X} m_{\x} A_\mathfrak{p}. \] From equalities \eqref{eq:=0} and \eqref{eq:f_x(y)}, for each $\y \in X$, \[ m_{\y} f_{\y}(\y)= - \sum_{\y \ne \x \in X} m_{\x} f_{\x}(\y) \in \mathfrak{p}A_{\mathfrak{p}}M. \] Since $f_{\y}(\y) \in A_{\mathfrak{p}} \setminus \mathfrak{p}A_{\mathfrak{p}}$ from equality \eqref{eq:f_x(x)}, it follows that $f_{\y}(\y) \in A_{\mathfrak{p}}^\times$ and \[ m_{\y} = - \sum_{\y \ne \x \in X} m_{\x} f_{\x}(\y)(f_{\y}(\y))^{-1} \in \mathfrak{p}A_{\mathfrak{p}}M. \] This implies that $M \subset \mathfrak{p}A_{\mathfrak{p}}M$, and hence $M = \mathfrak{p}A_{\mathfrak{p}}M$. By Nakayama's lemma, $M=\{0\}$ and $m_{\x}=0$ for each $ \x \in X$. Therefore $\{f_{\x}\}_{\x \in X}$ is linearly independent, and \[ |X| =|\{f_{\x}\}_{\x \in X}| \leq \dim P_s(\mathbb{R}^d)=\binom{d+s}{s} +\binom{d+s-1}{s-1} \] as desired. \end{proof} \begin{corollary} \label{coro:modp} Let $X$ be a subset of $\mathbb{R}^d$, and $\mathcal{O}_K$ the ring of integers of an algebraic number field $K$. Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$. Suppose $D(X) \subset\mathcal{O}_K$. If there exist $a_1,\ldots, a_s \in \mathcal{O}_K$ distinct modulo $\mathfrak{p}$ such that \begin{enumerate} \item for each $i \in \{1,\ldots,s\}$, $a_i \not \equiv 0 \pmod{\mathfrak{p}}$ and \item for each $\alpha \in D(X)$, there exists $i \in \{1,\ldots,s\}$ such that $\alpha \equiv a_i \pmod{\mathfrak{p}}$, \end{enumerate} then \[ |X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}. \] \end{corollary} \begin{proof} Since $A=\mathcal{O}_K \subset A_{\mathfrak{p}}$ and $A/\mathfrak{p} \cong A_\mathfrak{p}/\mathfrak{p} A_{\mathfrak{p}}$, this corollary is immediate from Theorem~\ref{thm:main}. \end{proof} \begin{example} For $X=\{(0,0),(1,0), (-\sqrt{3}/2,1/2),(-\sqrt{3}/2,-1/2)\} \subset \mathbb{R}^2$, the squared distances are $D(X)=\{1,2+\sqrt{3}\}$. We take the algebraic number field $K=\mathbb{Q}(\sqrt{3})$. Then the ring of integers is $\mathcal{O}_K=\mathbb{Z}+\sqrt{3}\mathbb{Z}$, and $\mathfrak{p}=(1+\sqrt{3})$ is a prime ideal of $\mathcal{O}_K$. Since $1\equiv 2+\sqrt{3} \pmod{\mathfrak{p}}$ holds, we have $|X| \leq \binom{d+1}{1}+\binom{d}{0}=4$. The set $X$ is an example attaining the upper bound in Corollary~\ref{coro:modp}. \end{example} We can prove a similar theorem to Theorem~\ref{thm:main} for an ideal $I \subset \mathcal{O}_K$ which may not be prime as follows. \begin{theorem}\label{thm:any_ideal} Let $X$ be a subset of $\mathbb{R}^d$, and $A=\mathcal{O}_K$ the ring of integers of an algebraic number field $K$. Let $I$ be an ideal of $A$, and $I=\mathfrak{p}_1^{\lambda_1} \cdots \mathfrak{p}_r^{\lambda_r}$ the prime decomposition of $I$. Let $A_{I}=S^{-1}A=\{a/s \mid a \in A, s \in S\}$, where $S=\bigcup_{R \in (A/I)^\times} R$. Suppose $D(X) \subset A_{I}$. If there exist $a_1,\ldots, a_s \in A_I$ distinct modulo $I A_{I}$ such that \begin{enumerate} \item for each $i \in \{1,\ldots,s\}$, $a_i \in A_I^\times$ and \item for each $\alpha \in D(X)$, there exists $i \in \{1,\ldots,s\}$ such that $\alpha \equiv a_i \pmod{I A_{I}}$, \end{enumerate} then \[ |X| \leq \binom{d+s}{s}+\binom{d+s-1}{s-1}. \] \end{theorem} \begin{proof} The proof is similar to that of Theorem~\ref{thm:main}, but we use $\mathfrak{p}_1\cdots \mathfrak{p}_r A_I$ instead of $\mathfrak{p} A_\mathfrak{p}$ as the ideal that is contained in all maximal ideals in Nakayama's lemma. \end{proof} For Theorem \ref{thm:any_ideal}, we must choose squared distances $a_i$ from $A_I^\times$. Such distances $a_i$ can be expressed by $a_i=s_1/s_2$ for some $s_1,s_2 \in S=\bigcup_{R \in (A/I)^\times} R$. Since $S \subset \bigcup_{R \in (A/\mathfrak{p}_j)^\times} R$ for any $j$, the squared distances $a_i$ are also elements of $A_{\mathfrak{p}_j}^\times=A\setminus \mathfrak{p}_j$. The natural homomorphisms \begin{align*} A_I/IA_I &\rightarrow A_I/\mathfrak{p}_1^{\lambda_1} A_I \times \cdots \times A_I/ \mathfrak{p}_r^{\lambda_r}A_I\\ &\rightarrow A_I/\mathfrak{p}_1 A_I \times \cdots \times A_I/ \mathfrak{p}_r A_I\\ &\rightarrow A_{\mathfrak{p}_1}/\mathfrak{p}_1 A_{\mathfrak{p}_1} \times \cdots \times A_{\mathfrak{p}_r}/ \mathfrak{p}_r A_{\mathfrak{p}_r} \end{align*} imply that the number of squared distances distinct modulo $IA_I$ is greater than or equal to that modulo $\mathfrak{p}_i A_{\mathfrak{p}_i}$ for any $i\in \{1,\ldots, r\}$. Therefore, Theorem \ref{thm:main} corresponding to the prime-ideal version gives the strongest upper bound for any ideal under our condition. \section{LRS type theorem} \label{sec:4} We now generalize the LRS type theorem proved in \cite{N11} as follows. The absolute bound $|X|\leq \binom{d+s}{s}$ is improved by this generalization. \begin{theorem} \label{thm:LRS} Suppose $s\geq 2$. Let $X$ be an $s$-distance set in $\mathbb{R}^d$ and $N=\dim P_{s-1}(\mathbb{R}^d)=\binom{d+s-1}{s-1}+\binom{d+s-2}{s-2}$. If $|X| \geq N+(N+1)/t$ for some $t \in \mathbb{N}$, then \begin{equation*} \label{eq:K_j} K_j=\prod_{i=1, i\ne j}^s \frac{\alpha_i}{\alpha_i-\alpha_j} \end{equation*} is an algebraic integer of degree at most $t$ for each $j \in \{1,\ldots,s \}$. \end{theorem} \begin{proof} Fix $j \in \{1,\ldots,s\}$. Define the polynomial \[ f(\x, \boldsymbol{\xi})=\prod_{i=1,i\ne j}^s \frac{\alpha_i-||\x-\boldsymbol{\xi}||^2}{\alpha_i-\alpha_j} \] for each $\x \in X$. Since $f(\x,\boldsymbol{\xi}) \in P_{s-1}(\mathbb{R}^d)$, the rank of the matrix $M=(f(\x,\y))_{\x,\y \in X}$ is at most $N$ \cite{N11}. The matrix can be expressed by \[ M=K_j I + A_j, \] where $I$ is the identity matrix and $A_j$ is a $(0,1)$-matrix with off diagonals. Since the size of $M$ is at least $N+(N+1)/t>N$, the matrix has $0$ eigenvalue whose multiplicity is at least $(N+1)/t$. This implies $-K_j$ is the eigenvalue of $A_j$, and hence $K_j$ is an algebraic integer. Assume $K_j$ is an algebraic integer of degree larger than $t$. Then the number of the conjugates of $-K_j$ is at least $t$, and the conjugates are also eigenvalues of $A_j$. Since $A_j$ has the eigenvalue $-K_j$ with multiplicity at least $(N+1)/t$, the size of $A_j$ is at least $(t+1)(N+1)/t=N+1+(N+1)/t$, which contradicts our assumption. Therefore $K_j$ is an algebraic integer of degree at most $t$. \end{proof} For $t=1$, the values $K_j$ are integers under the condition in Theorem~\ref{thm:LRS}, which is the previous result proved in \cite{N11}. The following corollaries are immediate from Theorem~\ref{thm:LRS}. \begin{corollary} If $K_j$ is not an algebraic integer for some $j \in \{1,\ldots, s\}$, then $|X| \leq~N$. \end{corollary} \begin{corollary}\label{coro:imp} Suppose $K_j$ is an algebraic integer for each $j \in \{1,\ldots,s\}$. Let $t$ be the maximum value of the degrees of $K_j$. If $t>1$ holds, then $|X| < N+(N+1)/(t-1)$. \end{corollary} Corollary \ref{coro:imp} is an improvement of the absolute bound for $s$-distance sets with the LRS ratios. If there exist $\alpha_i,\alpha_j \in D(X)\subset \mathcal{O}_K$ such that $\alpha_i$ is congruent to $\alpha_j$ modulo some prime ideal $\mathfrak{p}$ and $\alpha \not \equiv 0 \pmod{\mathfrak{p}}$ for each $\alpha \in D(X)$, then the LRS ratio $K_j$ is not an algebraic integer. Indeed, if $K_j \in \mathcal{O}_K$, then \begin{equation} \label{eq:no} 0\equiv K_j\prod_{i\ne j}(\alpha_i-\alpha_j) = \prod_{i \ne j} \alpha_i \not \equiv 0 \pmod{\mathfrak{p}}, \end{equation} which is a contradiction. When $K_j$ is not an algebraic integer for some $j$, we obtain the bound $|X|\leq N$ by Theorem~\ref{thm:LRS}, and we may obtain a better bound depending on the number of the elements of $D(X)$ distinct modulo $\mathfrak{p}$. The results proved in this paper -- mod-$\mathfrak{p}$ bound and LRS type theorem-- are analogously obtained for the sphere $S^{d-1}$ \cite{DGS77}, several projective spaces \cite{DL98,L92}, or $Q$-polynomial association schemes \cite{D73,DL98}. For spherical case, the LRS type theorem with $\mathcal{O}_K=\mathbb{Z}$ is useful to determine largest spherical $s$-distance sets for $s=2,3$. In \cite{GY18,MN11}, several largest $s$-distance sets are determined by a computer assistance. The possibilities of choices of integers $K_i$ are finite, and we can take the finite choices of distances from $K_i$. Reducing the number of the possible distances is helpful to cut the computational cost by a computer. However, Equation~\eqref{eq:no} implies that it is impossible to reduce the choices of distances by our results. \begin{remark} Akihiro Munemasa, one of the editors of the journal, communicated to the author the following idea to prove Theorem \ref{thm:main} without the use of Nakayama's lemma. Let $f_{\x}(\boldsymbol{\xi})$ be the same as in the proof of Theorem~\ref{thm:main}. We consider the matrix $M=(f_{\x} (\y))_{\x,\y \in X}$, where $X$ satisfies the condition of the theorem. In order to prove the linear independence of $\{f_{\x} \}_{\x \in X}$, it suffices to show that the determinant of $M$ is non-zero. The entries of $M$ are elements of $A_\mathfrak{p}$, and $M$ is congruent to some diagonal matrix modulo $\mathfrak{p} A_{\mathfrak{p}}$ whose diagonal entries are units in $A_\mathfrak{p}$. The determinant $M$ is not congruent to 0 modulo $\mathfrak{p} A_{\mathfrak{p}}$, in particular, it is non-zero. \end{remark} \longthanks{The author thanks Akihiro Munemasa for providing the idea of an alternative proof of Theorem \ref{thm:main} as editor's comments. The author is supported by JSPS KAKENHI Grant Numbers 18K03396, 19K03445, 20K03527, and 22K03402. } %\nocite{*} \bibliographystyle{mersenne-plain} \bibliography{ALCO_Nozaki_716.bib} \end{document}