Multifractal structure of Bernoulli convolutions

TL;DR

This paper investigates the multifractal spectrum of biased Bernoulli convolutions, revealing that for typical parameters, the spectrum's level sets are often nonempty and have positive Hausdorff dimension, even when the measure is absolutely continuous.

Contribution

It provides a detailed analysis of the multifractal spectrum of Bernoulli convolutions for typical parameters, extending understanding of their geometric complexity.

Findings

The multifractal spectrum's level sets are nonempty for many parameters.

These sets often have positive Hausdorff dimension.

Results hold even when the measure is absolutely continuous.

Abstract

Let $\nu_\lambda^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of $\nu_\lambda^p$ for typical $\lambda$. Namely, we investigate the size of the sets \[ \Delta_{\lambda,p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log \nu_\lambda^p(B(x,r))}{\log r} =\alpha\right\}. \] Our main results highlight the fact that for almost all, and in some cases all, $\lambda$ in an appropriate range, $\Delta_{\lambda,p}(\alpha)$ is nonempty and, moreover, has positive Hausdorff dimension, for many values of $\alpha$. This happens even in parameter regions for which $\nu_\lambda^p$ is typically absolutely continuous.