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

DOI:
https://doi.org/10.1090/S0002-9939-98-04460-8

MathSciNet review:
1458276

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Abstract: Since the 1930's many authors have studied the distribution of the random series where the signs are chosen independently with probability and . Solomyak recently proved that for almost every the distribution is absolutely continuous with respect to Lebesgue measure. In this paper we prove that is even equivalent to Lebesgue measure for almost all .

**1.**P.Erd\H{o}s (1939). On a family of symmetric Bernoulli convolutions,*Amer. J. Math.***61**, 974-976. MR**1:52a****2.**P.Erd\H{o}s (1940). On the smoothness properties of a family of Bernoulli convolutions,*Amer. J. Math.***62**, 180-186. MR**1:139e****3.**A.M. Garsia (1962). Arithmetic properties of Bernoulli convolutions,*Trans. Amer. Math. Soc.***102**, 409-432. MR**25:1409****4.**Y. Peres and B. Solomyak (1996a). Absolute continuity of Bernoulli convolutions,*Math. Research Letters***3:2**, 231-239. MR**97f:28006****5.**Y. Peres and B. Solomyak (1996b). Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc., to appear.**6.**B. Solomyak (1995). On the random series (an Erd\H{o}s problem),*Annals of Math.***142**, 611-625. MR**97d:11125****7.**A. Wintner (1935). On convergent Poisson convolutions,*Amer. J. Math.***57**, 827-838.

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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:
https://doi.org/10.1090/S0002-9939-98-04460-8

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