The author presents a survey on Bernoulli convolutions, based on a series of lectures given at the School on Real Analysis and Measure Theory, in Grado, 2001.
For a fixed \(\lambda\in(0,1)\) one defines a random series \(Y_\lambda=\sum_{n=0}^\infty\pm \lambda^n\), where the signs \(+\) or \(-\) are chosen independently, with probability \(1/2\). Its distribution \(\nu_\lambda\) is an infinite convolution product of \({{1}\over{2}}(\delta_{-\lambda^n}+ \delta_{\lambda^n})\), which gives its name of infinite Bernoulli convolutions. Bernoulli convolutions \(\nu_\lambda\) are self-similar measures associated to the IFS of contractive similitudes \(S_1, S_2:\mathbb{R}\to\mathbb{R}\), \(S_1(x)=\lambda x+1\), \(S_2(x)=\lambda x-1\), with the uniform probability vector \(p=(1/2, 1/2)\). An old result due to B. Jessen and A. Wintner [Trans. Am. Math. Soc. 38, 48–88 (1935; Zbl 0014.15401)] states that depending on \(\lambda\), the measure \( \nu_\lambda\) is either absolutely continuous with respect to the Lebesgue measure or purely singular. A natural question is the following: for which values \(\lambda\) is the measure \( \nu_\lambda\) absolutely continuous, and for which it is singular? By a result of R. Kershner and A. Wintner [Am. J. Math. 57, 541–548 (1935; Zbl 0012.06302)] the measure is singular for \(\lambda<1/2\).
\(\nu_{1/2}\) is the uniform measure on the interval \([-2,2]\). In 1939, Erdős proved that for \(\lambda\in(1/2,1)\) and \(1/\lambda\) a Pisot number (an algebraic integer greater than 1, whose all Galois conjugates have modulus less than one) the measure \(\nu_\lambda\) is singular. It remained an open problem whether for \(\lambda\in(1/2,1)\), and \(1/\lambda\) not a Pisot number, the measure \(\nu_\lambda\) is an absolute continuous measure or not. A first answer was given by Erdős in 1940. Namely, if \(S_{\perp}\) is the set of \(\lambda\in(1/2,1)\), such that \(\nu_\lambda\) is singular, then there exists \(a_0<1\) such that \(S_\perp\cap(a_0,1)\) has Lebesgue measure zero, and moreover, there exists an increasing sequence \((a_k)\) converging to 1, such that the measure \(\nu_\lambda\) has a density in \(C^k(\mathbb{R})\), for almost every \(\lambda\in(a_k,1)\).
The history of absolute continuity results on \(\nu_\lambda\), with \(\lambda\in(1/2,1)\), is discussed, and a gallery of histograms for \(\nu_\lambda\)-approximations is illustrated. The contribution of the author to the problem of absolute continuity of the measure \(\nu_\lambda\), for \(\lambda\in (1/2,1)\), appeared in [Ann. Math. (2) 142, No. 3, 611–625 (1995; Zbl 0837.28007)] and is rediscussed in this paper.
For the entire collection see [Zbl 1055.37002].