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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\lambda \in [\frac {1}{2},1]$.

- P.Erdős (1939). On a family of symmetric Bernoulli convolutions,
*Amer. J. Math.***61**, 974-976. - P.Erdős (1940). On the smoothness properties of a family of Bernoulli convolutions,
*Amer. J. Math.***62**, 180-186. - Adriano M. Garsia,
*Arithmetic properties of Bernoulli convolutions*, Trans. Amer. Math. Soc.**102**(1962), 409–432. MR**137961**, DOI https://doi.org/10.1090/S0002-9947-1962-0137961-5 - Yuval Peres and Boris Solomyak,
*Absolute continuity of Bernoulli convolutions, a simple proof*, Math. Res. Lett.**3**(1996), no. 2, 231–239. MR**1386842**, DOI https://doi.org/10.4310/MRL.1996.v3.n2.a8 - Y. Peres and B. Solomyak (1996b). Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc., to appear.
- Boris Solomyak,
*On the random series $\sum \pm \lambda ^n$ (an Erdős problem)*, Ann. of Math. (2)**142**(1995), no. 3, 611–625. MR**1356783**, DOI https://doi.org/10.2307/2118556 - A. Wintner (1935). On convergent Poisson convolutions,
*Amer. J. Math.***57**, 827–838.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
26A30,
28A78,
28A80

Retrieve articles in all journals with MSC (1991): 26A30, 28A78, 28A80

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

MR Author ID:
250279

Email:
matsimon@gold.uni-miskolc.hu

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