Department of Mathematics

Combinatorics and Graph Theory

  •  New bounds for equiangular lines and spherical two-distance sets
  •  01/31/2017
  •  4:10 PM - 5:00 PM
  •  C304 Wells Hall
  •  Wei-Hsuan Yu, MSU

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 {alpha, -alpha}, alpha in [0,1), are called equiangular. The problem of determining the maximal size of s-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an s-distance set in 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 is n(n+1)/2 with possible exceptions for some n = (2k+1)^2-3, k a positive integer. We also prove the universal upper bound ~ 2 n a^2/3 for equiangular sets with alpha = 1/a and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.



Department of Mathematics
Michigan State University
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C212 Wells Hall
East Lansing, MI 48824

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