Multifractal analysis of Bernoulli convolutions associated with Salem numbers

We consider the multifractal structure of the Bernoulli convolution ${\nu }_{\lambda }$, where ${\lambda }^{-1}$ is a Salem number in $(1,2)$. Let $\tau (q)$ denote the $L^q$ spectrum of ${\nu }_{\lambda }$. We show that if $\alpha \in [\tau’(+\infty), \tau’(0+)]$,thenthelevelset$$E(\alpha):={x\in \R:; \lim_{r\to 0}\frac{\log \nu_\lambda([x-r, x+r])}{\log r}=\alpha}$$ is non-empty and ${\dim }_{H}E(\alpha )={\tau }^{\ast }(\alpha )$, where ${\tau }^{\ast }$ denotes the Legendre transform of $\tau$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[{\tau }^{\mathrm{\prime }}(+\mathrm{\infty }),{\tau }^{\mathrm{\prime }}(0+)]$ is not a singleton when ${\lambda }^{-1}$ is the largest real root of the polynomial $x^n-x^{n-1}-\mathrm{.}\mathrm{.}\mathrm{.}-x+1$, $n\ge 4$. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.