Bernoulli convolutions -- 2023

Abstract

Let $θ\in(1,2)$, and $μ_θ$ be the Bernoulli convolution parametrized by $θ$, that is, the measure corresponding to the distribution of the random variable $\sum_{n=1}^{\infty} a_nθ^{-n}$, where the $a_n$ are i.i.d. with probability of $a_n=0$ equal to $\frac12$. As is well known, $μ_θ$ is either equivalent to the Lebesgue measure on $\text{supp}(μ_θ)$, or singular. Recall that an algebraic integer $>1$ is called Pisot if all its other Galois conjugates are smaller than 1 in modulus. It is known that $μ_θ$ is singular with $\dimμ_θ<1$ if $θ$ is Pisot. An algebraic integer $θ$ greater than 1 is called a Salem number if all its other Galois conjugates are of modulus 1, except $θ^{-1}$. I shall prove that (1) $\dimμ_θ=1$ if $θ$ is an algebraic non-Pisot number. (2) if $θ$ is Salem, then $μ_θ$ is equivalent to the Lebesgue measure on $\text{supp}(μ_θ)$, with an unbounded density in $L^p(\text{supp}(μ_θ))$ for all $p<\infty$. (3) Define \[ β_{θ,x,n}=\#\left\{a_1\dots a_n: \exists a_{n+1}\dots\text{such that\ } x=\sum_{k=1}^{\infty}a_nθ^{-k}\right\}. \] Then \[ \lim_{n\to\infty}\sqrt[n]{β_{θ,x,n}}=θ^{\dimμ_θ}\text{\ for}\ μ_θ-\text{a.e.} x. \] (4) Put \[ \bigcup_{n=1}^\infty\left\{\sum_{k=1}^{n}a_kθ^k\mid a_k\in\{-1,0,1\}\right\}= \{y_0(θ)<y_1(θ)<\cdots\}, \] and \[ \ell(θ)=\liminf_{n\to\infty}(y_{n+1}(θ)-y_n(θ)). \] I shall present a short proof of De-Jun Feng's famous theorem which states that $\ell(θ)=0$ for all non-Pisot $θ$.