Bounds for sets with few distances distinct modulo a prime ideal

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


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 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 [9], sets of equiangular lines in Euclidean spaces [14], and several s-distance sets in the real, complex, or quaternionic projective spaces [17] or in polynomial association schemes [5,6,7].
In the literature of combinatorial geometry, upper bounds for s-distance sets for Lintersecting families have been obtained. An L-intersecting family F is a family of subsets of a finite set F that satisfies |A ∩ B| ∈ L for any distinct A, B ∈ F for some L ⊂ {0, 1, . . . , n − 1}, where |F | = n. An L-intersecting family F is said to be k-uniform if |A| = k for each A ∈ F for some constant k. For k-uniform L-intersecting families F, Ray-Chaudhuri and Wilson [21] proved an upper bound |F| ≤ n s , where |L| = s. This case corresponds to s-distance sets in Johnson association schemes. After this work, Frankl and Wilson [8] obtained |F| ≤ s i=0 n i without the assumption of k-uniform.
Frankl and Wilson [8] also proved a modular version of the upper bound for k-uniform L-intersecting families. Namely they interpreted the sizes of the intersections as elements of Z/pZ 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 F, if k ≡ a (mod p) for each a ∈ L, then Frankl-Wilson [8] showed that |F| ≤ n s . Note that this is the same upper bound as that obtained under the assumption |L| = s. For L-intersecting families F with r different sizes of elements of F modulo p, Alon, Babai, and Suzuki [1] proved that |F| ≤ r−1 i=0 n s−i under a certain weak assumption which is simplified by [13]. The upper bound in [1] is proved by Koornwinder's method [15], 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 R d . For X in the unit sphere S d−1 , which corresponds to the condition of k-uniform, Delsarte, Goethals, and Seidel [6] proved that |X| ≤ d+s−1 s + d+s−2 s−1 . With no assumption, Bannai, Bannai, and Stanton [2] proved that |X| ≤ d+s s . For X in r concentric spheres, which corresponds to the condition of r different sizes, Bannai, Kawasaki, Nitamizu, and Sato [3] Recently simple alternative proofs of these upper bounds are given in [12,20].
Blokhuis [4] gave a modular version of upper bounds for Euclidean sets, assuming the squared distances are rational integers. Let D(X) be the set of the squared Euclidean distances between two distinct points of X. Theorem 1.1 (mod-p bound [4]). Let X be a subset of R d , and p a prime number. Suppose D(X) is a subset of rational integers Z. If there exist a 1 , . . . , a s ∈ Z distinct modulo p such that (1) for each i ∈ {1, . . . , s}, a i ≡ 0 (mod p) and (2) for each α ∈ D(X), there exists i ∈ {1, . . . , s} such that α ≡ a i (mod p), For sets in a sphere, several projective spaces, or Q-polynomial association schemes, Theorem 1.1 can be analogously obtained. However, the r-concentric spherical version is still open. The assumption D(X) ⊂ Z in Theorem 1.1 is surely strong, and the sets to which the theorem can be applied are restricted. In this paper, we extend Theorem 1.1 to the ring of integers O K of an algebraic number field K, and any prime ideal p of it. Note that throughout this paper, we fix an embedding of K into C, and K is interpreted as a subfield of C. We use Koornwinder's method to prove this mod-p upper bound. However the method to prove the linear independence of polynomial functions is new. In the proof, the localization A p of A = O K by a prime ideal p plays a key role and Nakayama's lemma is applied for a certain finitely generated A p -module. This method is purely algebraic and uniformly applicable to the polynomial spaces [10], [11,, which includes the Euclidean sphere, the real, complex or quaternionic projective spaces (see [7,17] 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) [5].
The paper is organized as follows. In Section 2, we introduce basic terminology and results about algebraic number fields. In Section 3, we prove a generalization of Theorem 1.1 (mod-p bound) for the ring of integers O K of an algebraic number field K and a prime ideal p ⊂ O K .
We also comment on the version of the theorem for an ideal that may not be prime. In Section 4, we extend the LRS type theorem proved in [16,19]. 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 4.1). We explain the relationship between the LRS type theorem and mod-p bound, which can refine an upper bound on the size of an s-distance set with given distances.

Preliminaries
An extension field K of rationals Q is an algebraic number field if the degree [K : Q] is finite. An algebraic number field K can be embedded into C, and K is always identified with a fixed specific subfield of C. The ring of integers O K is the ring consisting of all algebraic integers in K, where an 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 Suppose a ring is commutative and contains the identity. For a ring A, (A, m) is a local ring if A has a unique maximal ideal m. It is well known that for a ring A and its maximal ideal p ⊂ A, we can construct a local ring (A p , pA p ). For A = O K , the local ring is where S = A \ p. Its unique maximal ideal is pA p , which is the ideal of A p generated by the elements of p. Note that A p is a principal ideal domain. The natural map f : A/p → A p /pA 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, m), the ideal I in Theorem 2.1 is m. In order to prove the main theorem, we use the polynomial for x ∈ R d , a i ∈ R, and variables ξ = (ξ 1 , . . . , ξ d ), where ||x|| is the Euclidean norm of x. By proving the linear independence of {f x } x∈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 [2]. For ξ 0 = ξ 2 1 + · · · + ξ 2 d , we define the polynomial space P s (R d ) that consists of all real polynomial functions on R d which are spanned by ξ λ 0

Bounds for s-distance sets modulo p
The following is the main theorem in this paper.
Theorem 3.1 (mod-p bound). Let X be a subset of R d , and A = O K the ring of integers of an algebraic number field K. Let p be a prime ideal of A. Suppose D(X) ⊂ A p . If there exist a 1 , . . . , a s ∈ A p distinct modulo pA p such that (1) for each i ∈ {1, . . . , s}, a i ≡ 0 (mod pA p ) and (2) for each α ∈ D(X), there exists i ∈ {1, . . . , s} such that α ≡ a i (mod pA p ), Proof. For each x ∈ X, we define the polynomial f x (ξ) ∈ P s (R d ) as where if needed, we replace a i with a real value equivalent to a i modulo pA p . These polynomials satisfy and We prove {f x } x∈X is linearly independent as polynomial functions on R d . Assume there exist m x ∈ R such that x∈X m x f x (ξ) = 0. Since f y (y) ∈ A p \ pA p from equality (3.1), it follows that f y (y) ∈ A × p and This implies that M ⊂ pA p M , and hence M = pA p M . By Nakayama's lemma, M = {0} and m x = 0 for each x ∈ X. Therefore {f x } x∈X is linearly independent, and (2) for each α ∈ D(X), there exists i ∈ {1, . . . , s} such that α ≡ a i (mod p), Proof. Since A = O K ⊂ A p and A/p ∼ = A p /pA p , this corollary is immediate from Theorem 3.1.
We take the algebraic number field K = Q( √

3). Then the ring of integers is
The set X is an example attaining the upper bound in Corollary 3.2.
We can prove a similar theorem to Theorem 3.1 for an ideal I ⊂ O K which may not be prime as follows.
Proof. The proof is similar to that of Theorem 3.1, but we use p 1 · · · p r A I instead of pA p as the ideal that is contained in all maximal ideals in Nakayama's lemma.
For Theorem 3.4, we must choose squared distances a i from A × I . Such distances a i can be expressed by a i = s 1 /s 2 for some s 1 , s 2 ∈ S = R∈(A/I) × R. Since S ⊂ R∈(A/p j ) × R for any j, the squared distances a i are also elements of A × p j = A \ p j . The natural homomorphisms imply that the number of squared distances distinct modulo IA I is greater than or equal to that modulo p i A p i for any i ∈ {1, . . . , r}. Therefore, Theorem 3.1 corresponding to the prime-ideal version gives the strongest upper bound for any ideal under our condition.

LRS type theorem
We now generalize the LRS type theorem proved in [19] as follows. The absolute bound |X| ≤ d+s s is improved by this generalization.
is an algebraic integer of degree at most t for each j ∈ {1, . . . , s}.
Proof. Fix j ∈ {1, . . . , s}. Define the polynomial , the rank of the matrix M = (f (x, y)) x,y∈X is at most N [19]. The matrix can be expressed by 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.
For t = 1, the values K j are integers under the condition in Theorem 4.1, which is the previous result proved in [19]. The following corollaries are immediate from Theorem 4.1.  If there exist α i , α j ∈ D(X) ⊂ O K such that α i is congruent to α j modulo some prime ideal p and α ≡ 0 (mod p) for each α ∈ D(X), then the LRS ratio K j is not an algebraic integer. Indeed, if K j ∈ O K , then which is a contradiction. When K j is not an algebraic integer for some j, we obtain the bound |X| ≤ N by Theorem 4.1, and we may obtain a better bound depending on the number of the elements of D(X) distinct modulo p.
The results proved in this paper -mod-p bound and LRS type theorem-are analogously obtained for the sphere S d−1 [6], several projective spaces [7,17], or Q-polynomial association schemes [5,7]. For spherical case, the LRS type theorem with O K = Z is useful to determine largest spherical s-distance sets for s = 2, 3. In [9,18], 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 (4.1) implies that it is impossible to reduce the choices of distances by our results.
Remark 4.4. Akihiro Munemasa, one of the editors of the journal, communicated to the author the following idea to prove Theorem 3.1 without the use of Nakayama's lemma. Let f x (ξ) be the same as in the proof of Theorem 3.1. We consider the matrix M = (f x (y)) x,y∈X , where X satisfies the condition of the theorem. In order to prove the linear independence of {f x } x∈X , it suffices to show that the determinant of M is non-zero. The entries of M are elements of A p , and M is congruent to some diagonal matrix modulo pA p whose diagonal entries are units in A p . The determinant M is not congruent to 0 modulo pA p , in particular, it is non-zero.