Bernoulli convolution measures and their Fourier transforms

Mon, 12/01/2015 - 14:00

For $\lambda\in (0,1)$, the Bernoulli convolution measure $\nu_\lambda$ may be defined as the distribution 
of the random series $\sum_{n=0}^\infty \pm \lambda^n$, where the signs are chosen independently with equal 
probabilities. For $\lambda =1/3$, this is the familiar Cantor-Lebesgue measure (up to a linear change of variable). 
The Fourier transform of $\nu_\lambda$ has an infinite product formula:
$$\widehat{\nu}_\lambda(t) = \prod_{n=0}^\infty \cos(2\pi \lam^n t).$$
The properties of $\nu_\lambda$ and their Fourier transforms have been studied since the 1930's by many mathematicians, 
among them Jessen, Wintner, Erd\H{o}s, Salem, Kahane, Garcia. In particular, it was proved by Erd\H{o}s and Salem that 
$\widehat{\nu}_\lambda(t)$ does not vanish at infinity (i.e. $\nu_\lambda$ is not a Rajchman measure) if and only if 
$1/\lambda$ is a Pisot number (an algebraic integer greater than one with all conjugates inside the unit circle). 
However, very little is known about the rate of decay, especially for specific $\lambda$, as opposed to "typical" ones. 
In this talk I will survey known results and open problems in this direction. Recently in a joint work with A. Bufetov 
we proved that if $1/\lam$ is an algebraic integer with at least one conjugate outside of the unit circle, then the 
Fourier transform of $\nu_\lam$ has at least a logarithmic decay rate at infinity.