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Bounds for sets with few distances distinct modulo a prime ideal
Nozaki, Hiroshi1
1 Aichi University of Education Department of Mathematics Education 1 Hirosawa, Igaya-cho Kariya, Aichi 448-8542 (Japan)
Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 539-545.

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)⊂𝒪 K and there exist s values a 1 ,...,a s ∈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|≤d+s s+d+s-1 s-1.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.272
Classification: 05D05, 05B30
Keywords: $s$-distance set, algebraic number field

Nozaki, Hiroshi 1

1 Aichi University of Education Department of Mathematics Education 1 Hirosawa, Igaya-cho Kariya, Aichi 448-8542 (Japan)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Nozaki, Hiroshi. Bounds for sets with few distances distinct modulo a prime ideal. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 539-545. doi : 10.5802/alco.272. https://alco.centre-mersenne.org/articles/10.5802/alco.272/

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