On relationships between shift radix systems and canonical number systems

De-Jun Feng

The Chinese University of Hong Kong, Shatin, Hong Kong, China

For $\lambda\in (1/2,1)$, the Bernoulli convolution measure associated with $\lambda$ is the probability distribution $\mu_\lambda$ of the random series $\sum_{n=0}^\infty \pm \lambda^n$, where the ``$+$'' and ``$-$'' signs are chosen independently with probability 1/2. Bernoulli convolutions have been studied since the beginning of the 20th century. It is an interesting fact that the arithmetic property of $\lambda$ will affect the analytic property of $\mu_\lambda$. In this talk I will mention some results about the local structure, the Hausdorff dimension and the smoothness of Bernoulli convolutions associated some concrete algebraic numbers $\lambda$.