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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.