Bernoulli convolutions -- 2023

Let $\theta \in (1,2)$, and ${\mu }_{\theta }$ be the Bernoulli convolution parametrized by $\theta$, that is, the measure corresponding to the distribution of the random variable ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n{\theta }^{-n}$, where the $a_n$ are i.i.d. with probability of $a_n=0$ equal to $\frac{1}{2}$. As is well known, ${\mu }_{\theta }$ is either equivalent to the Lebesgue measure on $\text{supp}({\mu }_{\theta })$, 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 ${\mu }_{\theta }$ is singular with $\dim {\mu }_{\theta }<1$ if $\theta$ is Pisot. An algebraic integer $\theta$ greater than 1 is called a Salem number if all its other Galois conjugates are of modulus 1, except ${\theta }^{-1}$. I shall prove that (1) $\dim {\mu }_{\theta }=1$ if $\theta$ is an algebraic non-Pisot number. (2) if $\theta$ is Salem, then ${\mu }_{\theta }$ is equivalent to the Lebesgue measure on $\text{supp}({\mu }_{\theta })$, with an unbounded density in $L^p(\text{supp}({\mu }_{\theta }))$ for all $p<\mathrm{\infty }$. (3) Define $${\beta }_{\theta ,x,n}=\mathrm{\#}\{a_1\dots a_n:\mathrm{\exists }{a}_{n+1}\dots \text{such that }x=\sum _{k=1}^{\mathrm{\infty }}{a}_n{\theta }^{-k}\}\mathrm{.}$$ Then $$\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{{\beta }_{\theta ,x,n}}={\theta }^{\dim {\mu }_{\theta }}\text{ for }{\mu }_{\theta }-\text{a.e.}x\mathrm{.}$$ (4) Put $$\bigcup _{n=1}^{\mathrm{\infty }}\{\sum _{k=1}^n{a}_k{\theta }^k\mid a_k\in \{-1,0,1\}\}=\{y_0(\theta )<y_1(\theta )<\cdots \},$$ and $$\mathrm{\ell }(\theta )=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}(y_{n+1}(\theta )-y_n(\theta ))\mathrm{.}$$ I shall present a short proof of De-Jun Feng’s famous theorem which states that $\mathrm{\ell }(\theta )=0$ for all non-Pisot $\theta$.