Counting beta -expansions and the absolute continuity of Bernoulli convolutions

摘要

We study the typical growth rate of the number of words of length (n) which can be extended to (beta )-expansions of (x). In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter (beta ) is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.