Bernoulli convolutions associated with certain non-Pisot numbers

The Bernoulli convolution ν_λ measure is shown to be absolutely continuous with L² density for almost all , and singular if λ⁻¹ 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 ν_λ such that their density functions, if they exist, are not L². We also construct other Bernoulli convolutions whose density functions, if they exist, behave rather badly.