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Spherical sets avoiding a prescribed set of angles
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DeCorte, E. and Pikhurko, Oleg (2016) Spherical sets avoiding a prescribed set of angles. International Mathematics Research Notices, 2016 (20). pp. 6095-6117. doi:10.1093/imrn/rnv319 ISSN 1073-7928.
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Official URL: http://dx.doi.org/10.1093/imrn/rnv319
Abstract
Let X be any subset of the interval [−1, 1]. A subset I of the unit sphere in Rn will be called X-avoiding if u, v ∈/ X for any u, v ∈ I. The problem of determining the maximum surface measure of a {0}-avoiding set was first stated in a 1974 note by Witsenhausen; there the upper bound of 1/n times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl–Wilson theorem is that this fraction decreases exponentially, but until now the 1/3 upper bound for the case n= 3 has not moved. We improve this bound to 0.313 using an approach inspired by Delsarte’s linear programming bounds for codes, combined with some combinatorial reasoning. In the second part of the paper, we use harmonic analysis to show that, for n≥ 3, there always exists an X-avoiding set of maximum measure. We also show with an example that a maximizer need not exist when n= 2.
Item Type: | Journal Article | ||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||
Library of Congress Subject Headings (LCSH): | Spherical functions | ||||||||||
Journal or Publication Title: | International Mathematics Research Notices | ||||||||||
Publisher: | Oxford University Press | ||||||||||
ISSN: | 1073-7928 | ||||||||||
Official Date: | 1 January 2016 | ||||||||||
Dates: |
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Volume: | 2016 | ||||||||||
Number: | 20 | ||||||||||
Number of Pages: | 23 | ||||||||||
Page Range: | pp. 6095-6117 | ||||||||||
DOI: | 10.1093/imrn/rnv319 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||||
Date of first compliant deposit: | 28 January 2016 | ||||||||||
Date of first compliant Open Access: | 1 March 2016 | ||||||||||
Funder: | European Research Council (ERC), Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Netherlands Organisation for Scientific Research] (NWO), Lady Davis Fellowship Trust, Engineering and Physical Sciences Research Council (EPSRC) | ||||||||||
Grant number: | 320924-ProGeoCom (ERC), 306493-EC (ERC), 639.032.917 (NWO), EP/K012045/1 (EPSRC) |
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