Upper bounds for s-distance sets and equiangular lines
| Autor: | Wei-Hsuan Yu, Alexey Glazyrin |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Semidefinite programming
General Mathematics 010102 general mathematics Scalar (mathematics) Equiangular polygon Metric Geometry (math.MG) 0102 computer and information sciences 01 natural sciences Upper and lower bounds Combinatorics Metric space Mathematics - Metric Geometry 010201 computation theory & mathematics Bounding overwatch FOS: Mathematics Mathematics - Combinatorics Pairwise comparison Combinatorics (math.CO) 0101 mathematics Equiangular lines Mathematics |
| Zdroj: | Advances in Mathematics. 330:810-833 |
| ISSN: | 0001-8708 |
| DOI: | 10.1016/j.aim.2018.03.024 |
| Popis: | The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products { α , − α } , α ∈ [ 0 , 1 ) , are called equiangular. The problem of determining the maximum size of s-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an s-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in R n , n ≥ 7 , is n ( n + 1 ) 2 with possible exceptions for some n = ( 2 k + 1 ) 2 − 3 , k ∈ N . We also prove the universal upper bound ∼ 2 3 n a 2 for equiangular sets with α = 1 a and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound. |
| Databáze: | OpenAIRE |
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