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 \[ \dim (μ_β) \geq 0.9804085,\] 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 ${\beta }_{\ast }$ in a small interval \[ (β_{3} -10^{-8}, β_{3} + 10^{-8}).\] 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.

Some minor changes and clarifications. To appear in J. Lond. Math. Soc