The dimension of random subsets of self-similar sets generated by branching random walk

Abstract

Given a self-similar set \(\Lambda\) that is the attractor of an iterated function system (IFS) \(\{f_1,\dots,f_N\}\), consider the following method for constructing a random subset of \(\Lambda\): Let \(\mathbf{p}=(p_1,\dots,p_N)\) be a probability vector, and label all edges of a full \(M\)-ary tree independently at random with a number from \(\{1,2,\dots,N\}\) according to \(\mathbf{p}\), where \(M\geq 2\) is an arbitrary integer. Then each infinite path in the tree starting from the root receives a random label sequence which is the coding of a point in \(\Lambda\). We let \(F\subset\Lambda\) denote the set of all points obtained in this way. This construction was introduced by Allaart and Jones (2025), who considered the case of a homogeneous IFS on \(\mathbb{R}\) satisfying the Open Set Condition (OSC) and proved non-trivial upper and lower bounds for the Hausdorff dimension of \(F\). We demonstrate that under the OSC, the Hausdorff (and box-counting) dimension of \(F\) is equal to the upper bound of Allaart and Jones, and extend the result to higher dimensions as well as to non-homogeneous self-similar sets.