Rates of convergence towards the boundary of a self-similar set

The geometry of self-similar sets $K$ has been studied intensively during the past 20 years frequently assuming the so-called Open Set Condition (OSC). The OSC guarantees the existence of an open set $U$ satisfying various natural invariance properties, and is instrumental in the study of self-similar sets for the following reason: a careful analysis of the boundaries of the iterates of $\overline U$ is the key technique for obtaining information about the geometry of $K$. In order to obtain a better understanding of the OSC and because of the geometric significance of the boundaries of the iterates of $\overline U$, it is clearly of interest to provide quantitative estimates for the "number" of points close to the boundaries of the iterates of $\overline U$. This motivates a detailed study of the rate at which the distance between a point in $K$ and the boundaries of the iterates of $\overline U$ converge to $0$. In this paper we show that for each $t\in I$ (where $I$ is a certain interval defined below) there is a significant number of points for which the rate of convergence equals $t$. In fact, for each $t\in I$, we show that the set of points whose rate of convergence equals $t$ has positive Hausdorff dimension, and we obtain a lower bound for this dimension. Examples show that this bound is, in general, the best possible and cannot be improved.