TY - JOUR

T1 - Upper bounds for s-distance sets and equiangular lines

AU - Glazyrin, Alexey

AU - Yu, Wei Hsuan

N1 - Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/5/25

Y1 - 2018/5/25

N2 - 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 Rn, n≥7, is [Formula presented] with possible exceptions for some n=(2k+1)2−3, k∈N. We also prove the universal upper bound ∼[Formula presented]na2 for equiangular sets with α=[Formula presented] 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.

AB - 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 Rn, n≥7, is [Formula presented] with possible exceptions for some n=(2k+1)2−3, k∈N. We also prove the universal upper bound ∼[Formula presented]na2 for equiangular sets with α=[Formula presented] 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.

KW - Equiangular lines

KW - s-distance sets

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=85044846581&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.03.024

DO - 10.1016/j.aim.2018.03.024

M3 - 期刊論文

AN - SCOPUS:85044846581

VL - 330

SP - 810

EP - 833

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -