The Bernoulli convolution ν λ measure is shown to be absolutely continuous with L 2 density for almost all 12<λ<1,and singular if λ -1 is a Pisot number. It is an open question whether the Pisot typ...The Bernoulli convolution ν λ measure is shown to be absolutely continuous with L 2 density for almost all 12<λ<1,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 L 2. We also construct other Bernolulli convolutions whose density functions,if they exist,behave rather badly.展开更多
文摘The Bernoulli convolution ν λ measure is shown to be absolutely continuous with L 2 density for almost all 12<λ<1,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 L 2. We also construct other Bernolulli convolutions whose density functions,if they exist,behave rather badly.
