Laws of pure type 4 . Dimensions of Bernoulli convolutions 5 . Bernoulli convolutions and Salem numbers 6

The distribution νλ of the random series ∑ ±λ is the infinite convolution product of 1 2 (δ−λn + δλn). These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which λ ∈ ( 1 2 , 1) is νλ absolutely continuous? Our main goal is to present an exposition of results obtained by Erdős, Kahane and the authors on this problem. Several related unsolved problems are collected at the end of the paper.

[1]  M. Keane,et al.  ON THE MORPHOLOGY OF y-EXPANSIONS WITH DELETED DIGITS , 2005 .

[2]  Vladimir I. Clue.  Harmonic analysis , 2004, 2004 IEEE Electro/Information Technology Conference.

[3]  P-4ur,et al.  ON A FAMILY OF SYMMETRIC BERNOULLI CONVOLUTIONS , 2002 .

[4]  Y. Peres,et al.  Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions , 2000 .

[5]  B. Solomyak.  Measure and dimension for some fractal families , 1998, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  S. Lalley.  Random Series in Powers of Algebraic Integers: Hausdorff Dimension of the Limit Distribution , 1998 .

[7]  R. Mauldin,et al.  The equivalence of some Bernoulli convolutions to Lebesgue measure , 1998 .

[8]  Y. Peres,et al.  Self-similar measures and intersections of Cantor sets , 1998 .

[9]  K. Lau,et al.  $L^q$-spectrum of the Bernoulli convolution associated with the golden ratio , 1998 .

[10]  J. Yorke,et al.  Are the dimensions of a set and its image equal under typical smooth functions? , 1997, Ergodic Theory and Dynamical Systems.

[11]  B. Solomyak.  On the measure of arithmetic sums of Cantor sets , 1997 .

[12]  The local dimensions of the Bernoulli convolution associated with the golden number , 1997 .

[13]  A. Vershik,et al.  Ergodic properties of Erd\"os measure, the entropy of the goldenshift, and related problems , 1996, math/9612223.

[14]  Spatially chaotic configurations and functional equations with rescaling , 1996 .

[15]  Y. Peres,et al.  Absolute Continuity of Bernoulli Convolutions, A Simple Proof , 1996 .

[16]  Russell Lyons,et al.  Seventy Years of Rajchman Measures , 2020 .

[17]  K. Simon,et al.  The Hausdorff dimension of -expansions with deleted digits , 1995 .

[18]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[19]  Tong Zhang,et al.  Densities of Self-Similar Measures on the Line , 1995, Exp. Math..

[20]  Jonathan M. Borwein,et al.  Functional Equations and Distribution Functions , 1994 .

[21]  F. Ledrappier,et al.  A dimension formula for Bernoulli convolutions , 1994 .

[22]  K. Lau.  Dimension of a Family of Singular Bernoulli Convolutions , 1993 .

[23]  F. Ledrappier.  On the dimension of some graphs , 1992 .

[24]  J. Alexander,et al.  The Entropy of a Certain Infinitely Convolved Bernoulli Measure , 1991 .

[25]  G. Weiss,et al.  Littlewood-Paley Theory and the Study of Function Spaces , 1991 .

[26]  P. Mattila.  ORTHOGONAL PROJECTIONS, RIESZ CAPACITIES, AND MINKOWSKI CONTENT , 1990 .

[27]  M. Urbanski,et al.  On the Hausdorff dimension of some fractal sets , 1989 .

[28]  Kenneth Falconer,et al.  The Hausdorff dimension of self-affine fractals , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[29]  James A. Yorke,et al.  Fat baker's transformations , 1984, Ergodic Theory and Dynamical Systems.

[30]  K. Falconer.  Hausdorff dimension and the exceptional set of projections , 1982 .

[31]  L. Young.  Dimension, entropy and Lyapunov exponents , 1982, Ergodic Theory and Dynamical Systems.

[32]  David W. Boyd,et al.  Speculations Concerning the Range of Mahler's Measure , 1980, Canadian Mathematical Bulletin.

[33]  Pertti Mattila,et al.  Hausdorff dimension, orthogonal projections and intersections with planes , 1975 .

[34]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[35]  Yves Meyer,et al.  Algebraic numbers and harmonic analysis , 1972 .

[36]  Sur la distribution de certaines séries aléatoires , 1971 .

[37]  D. Boyd.  TRANSCENDENTAL NUMBERS WITH BADLY DISTRIBUTED POWERS , 1969 .

[38]  R. Kaufman.  On Hausdorff dimension of projections , 1968 .

[39]  A. Garsia.  Entropy and singularity of infinite convolutions. , 1963 .

[40]  R. Salem.  Algebraic numbers and Fourier analysis , 1963 .

[41]  A. Garsia.  Arithmetic properties of Bernoulli convolutions , 1962 .

[42]  Jean Kahane,et al.  Sur la convolution d'une infinité de distributions de Bernoulli , 1958 .

[43]  J. M. Marstrand.  Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions , 1954 .

[44]  R. Salem,et al.  A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan , 1944 .

[45]  On Convergent Poisson Convolutions , 1935 .

[46]  On Symmetric Bernoulli Convolutions , 1935 .

[47]  A. Wintner,et al.  Distribution functions and the Riemann zeta function , 1935 .