RÉSUMÉ

Let s > 1 be our parameter. The Bernoulli convolution $\mu_s$ parameterised by s is the infinite convolution of the two-point measures, the n-th of which is supported on $+s^{-n}$ and $-s^{-n}$ with probabilities 1/2. As was shown by R. Salem, unless s is a Pisot number, the set of limit points of $\mu_s$ at the infinity, $J_s$, is simply {0}. On the other hand, it was shown by P. Erdos (1939) that if s is Pisot, $J_s$ contains nonzero points. In 1980 P. Sarnak, in connection with an important question regarding the spectrum of the convolution operator for $\mu_s$, asked whether $J_s$ is countable in this case and showed that if s is an integer, the answer is yes. In the present talk we will give a sketch of the proof that the answer is positive for each Pisot s. Moreover, each derived set of $J_s$ proves to be infinite countable as well. The techniques used are typical for the theory of Pisot numbers. This work is joint with B. Solomyak.

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