On the dimension of Bernoulli convolutions
dc.contributor.author | Breuillard, Emmanuel | |
dc.contributor.author | Varjú, PP | |
dc.date.accessioned | 2019-10-11T23:30:53Z | |
dc.date.available | 2019-10-11T23:30:53Z | |
dc.date.issued | 2019-07 | |
dc.identifier.issn | 0091-1798 | |
dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/297783 | |
dc.description.abstract | The Bernoulli convolution with parameter λ ∈ (0, 1) is the probability measure μλ that is the law of the random variable σn ≥ 0 ±λn, where the signs are independent unbiased coin tosses. We prove that each parameter λ ∈ (1/2, 1) with dimμλ < 1 can be approximated by algebraic parameters η ∈ (1/2, 1) within an error of order exp(-deg(η)A) such that dimμη < 1, for any number A. As a corollary, we conclude that dimμλ = 1 for each of λ = ln 2, e-1/2,π/4. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant a < 1 such that dimμλ = 1 for all λ ∈ (a, 1). | |
dc.publisher | Institute of Mathematical Statistics | |
dc.rights | All rights reserved | |
dc.title | On the dimension of Bernoulli convolutions | |
dc.type | Article | |
prism.endingPage | 2617 | |
prism.issueIdentifier | 4 | |
prism.publicationName | Annals of Probability | |
prism.startingPage | 2582 | |
prism.volume | 47 | |
dc.identifier.doi | 10.17863/CAM.44836 | |
rioxxterms.versionofrecord | 10.1214/18-AOP1324 | |
rioxxterms.version | VoR | |
rioxxterms.licenseref.uri | http://www.rioxx.net/licenses/all-rights-reserved | |
dc.identifier.eissn | 2168-894X | |
rioxxterms.type | Journal Article/Review | |
pubs.funder-project-id | Royal Society (UF140146) | |
cam.issuedOnline | 2019-07-04 |
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