On the random series \(\sum\pm\lambda^ n\) (an Erdös problem). (English) Zbl 0837.28007

Solomyak, Boris

Ann. Math. (2) 142, No. 3, 611-625 (1995).

The author gives an important partial answer to an Erdös problem. He shows that the distribution \(\nu_\lambda\) of the random series \(Y_\lambda= \sum^\infty_{n= 0} \pm \lambda^n\) has a density in \(L^2(\mathbb{R})\) for Lebesgue-a.e. \(\lambda\in (0, 5, 1)\), where the sign is chosen with probability \({1\over 2}\). To this end he extends the series \(Y_\lambda\). Instead of two valued random variables he uses independent random variables taking values in a finite set. He must prove a general result concerning multiple zeros of classes of power series to apply it together with the extended versions of random series to get the above Erdös result.
Finally, he notes that the measure \(\nu_\lambda\) occurs as a component of the Bowen-Ruelle measure for the fat baker’s transformation as well as it is used to prove coincidence of dimensions for certain self-affine graphs.

Reviewer: H.Haase (Greifswald)

MSC:

28A80

Fractals

Keywords:

Bernoulli convolutions; Erdös problem; random series; power series; Bowen-Ruelle measure; baker’s transformation; dimensions; self-affine graphs