Estimates on the dimension of self‐similar measures with overlaps

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 β ∈ (1, 2), the dimension of the Bernoulli convolution μβ satisfies dim(μβ) ≥ 0.98040856, which improves a previous uniform lower bound 0.82 obtained by Hare and Sidorov [16]. This new uniform lower bound is very close to the known numerical approximation 0.98040931953±10−11 for dimμβ3 , where β3 ≈ 1.839286755214161 is the largest root of the polynomial x−x−x− 1. Moreover, the infimum infβ∈(1,2) dim(μβ) is attained at a parameter β∗ in a small interval (β3 − 10−8, β3 + 10−8). When β is a Pisot number, we express dim(μβ) in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate dim(μβ) from above as well.