Bounds for sets with few distances distinct modulo a prime ideal

Nozaki, Hiroshi

doi:10.5802/alco.272

Geodesic

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Bounds for sets with few distances distinct modulo a prime ideal

Nozaki, Hiroshi ¹

¹ Aichi University of Education Department of Mathematics Education 1 Hirosawa, Igaya-cho Kariya, Aichi 448-8542 (Japan)

Algebraic Combinatorics, Tome 6 (2023) no. 2, pp. 539-545

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Résumé

Let $𝒪_{K}$ be the ring of integers of an algebraic number field $K$ embedded into $ℂ$ . Let $X$ be a subset of the Euclidean space $ℝ^{d}$ , and $D (X)$ be the set of the squared distances of two distinct points in $X$ . In this paper, we prove that if $D (X) \subset 𝒪_{K}$ and there exist $s$ values $a_{1}, ..., a_{s} \in 𝒪_{K}$ distinct modulo a prime ideal $𝔭$ of $𝒪_{K}$ such that each $a_{i}$ is not zero modulo $𝔭$ and each element of $D (X)$ is congruent to some $a_{i}$ , then $| X | \leq (\binom{d + s}{s}) + (\binom{d + s - 1}{s - 1})$ .

Reçu le : 2022-03-24
Révisé le : 2022-10-07
Accepté le : 2022-10-10
Publié le : 2023-05-03

DOI : 10.5802/alco.272

Classification : 05D05, 05B30
Keywords: $s$-distance set, algebraic number field

Affiliations des auteurs :

Nozaki, Hiroshi ¹

¹ Aichi University of Education Department of Mathematics Education 1 Hirosawa, Igaya-cho Kariya, Aichi 448-8542 (Japan)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Bounds for sets with few distances distinct modulo a prime ideal},
     journal = {Algebraic Combinatorics},
     pages = {539--545},
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     doi = {10.5802/alco.272},
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Nozaki, Hiroshi. Bounds for sets with few distances distinct modulo a prime ideal. Algebraic Combinatorics, Tome 6 (2023) no. 2, pp. 539-545. doi: 10.5802/alco.272

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