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On probability measures arising from lattice points on circles
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4734-5092
2016 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, 1-42 p.Article in journal (Refereed) PublishedText
Abstract [en]

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice (Formula presented.), gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a “fractal” structure. This complicated structure in some sense arises from prime powers—singularities do not occur for circles of radius (Formula presented.) if n is square free.

Place, publisher, year, edition, pages
2016. 1-42 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-186995DOI: 10.1007/s00208-016-1411-4OAI: oai:DiVA.org:kth-186995DiVA: diva2:929911
Note

QC 20160520

Available from: 2016-05-20 Created: 2016-05-16 Last updated: 2016-05-20Bibliographically approved

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Kurlberg, Pär
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