Bernoulli convolutions associated with certain non-Pisot numbers

Abstract

The Bernoulli convolution v<inf>λ</inf> measure is shown to be absolutely continuous with L² density for almost all 1/2 <λ<1, and singular if λ^-λ is a Pisot number. It is an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions Vi such that their density functions, if they exist, are not L². We also construct other Bernolulli convolutions whose density functions, if they exist, behave rather badly.