Estimates on the dimension of self-similar measures with overlaps

Abstract

In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any $ \beta\in(1,2) $, the dimension of the Bernoulli convolution $ \mu_\beta $ satisfies \[ \dim (\mu_\beta) \geq 0.9804085,\] which improves a previous uniform lower bound $0.82$ obtained by Hare and Sidorov \cite{HareSidorov2018}. This new uniform lower bound is very close to the known numerical approximation $ 0.98040931953\pm 10^{-11}$ for $\dim \mu_{\beta_3}$, where $ \beta_{3} \approx 1.839286755214161$ is the largest root of the polynomial $ x^{3}-x^{2}-x-1$. Moreover, the infimum $\inf_{\beta\in (1,2)}\dim (\mu_\beta) $ is attained at a parameter $\beta_*$ in a small interval \[ (\beta_{3} -10^{-8}, \beta_{3} + 10^{-8}).\] When $\beta$ is a Pisot number, we express $\dim(\mu_\beta)$ in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate $\dim (\mu_\beta)$ from above as well.