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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The equivalence of some Bernoulli
convolutions to Lebesgue measure


Authors: R. Daniel Mauldin and Károly Simon
Journal: Proc. Amer. Math. Soc. 126 (1998), 2733-2736
MSC (1991): Primary 26A30, 28A78, 28A80
MathSciNet review: 1458276
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Abstract: Since the 1930's many authors have studied the distribution $\nu _{\lambda}$ of the random series $Y_{\lambda}=\sum \pm {\lambda}^n$ where the signs are chosen independently with probability $(1/2,1/2)$ and $0<\lambda<1$. Solomyak recently proved that for almost every $\lambda\in [\frac{1}{2},1],$ the distribution $\nu _{\lambda}$ is absolutely continuous with respect to Lebesgue measure. In this paper we prove that $\nu _{\lambda}$ is even equivalent to Lebesgue measure for almost all $\lambda\in [\frac{1}{2},1]$.


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Additional Information

R. Daniel Mauldin
Affiliation: Department of Mathematics, P. O. Box 305118, University of North Texas, Denton, Texas 76203-5118
Email: mauldin@dynamics.math.unt.edu

Károly Simon
Affiliation: Department of Mathematics, P. O. Box 305118, University of North Texas, Denton, Texas 76203-5118
Address at time of publication: Institute of Mathematics, University of Miskolc, Miskolc-Egyetem- varos, H-3515 Hungary
Email: matsimon@gold.uni-miskolc.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04460-8
PII: S 0002-9939(98)04460-8
Keywords: Bernoulli convolution, equivalent measures
Received by editor(s): February 11, 1997
Additional Notes: The first author’s research was supported by NSF Grant DMS-9502952. The second author’s research was partially supported by grants F19099 and T19104 from the OTKA Foundation
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1998 American Mathematical Society