Random covering by rectangles on self-similar carpets
E Daviaud
TL;DR
This work analyzes random geometric approximation on two-dimensional self-similar carpets by studying random coverings with rectangles and anisotropic shrinking targets. It develops a multifractal- and ergodic-theoretic framework for base-$b$ missing-digit IFSs, linking the dimension of limsup sets to the projection spectrum $D_{\pi_2 \mu_0}$ and to local dimensions along the $y$-axis via $\alpha_\nu$. The authors establish piecewise-dimension formulas for the random rectangle covering problem and its tree (shrinking target) counterpart, with explicit expressions in terms of $\dim_H K$, $\alpha_\nu$, and the multifractal spectrum, and provide conditions under which these bounds are sharp. The results are extended to both i.i.d. samples from $\mu_0$ and to dynamical orbits under the $\times b$ map, using a mass transference principle and content estimates to derive sharp dimension statements. These findings illuminate anisotropic approximation on fractals and connect random covering phenomena to multifractal geometry and ergodic theory on self-similar sets.
Abstract
In this article, given a base-b self-similar set K, we study the random covering of K by horizontal or vertical rectangles, with respect to the Alfhors-regular measure on K, and the rectangular shrinking target problem on K.