Codimension formulae for the intersection of fractal subsets of Cantor spaces (Q2789863)

Let \(\mathcal{C}^m := \{1, 2, \ldots, m\}^\mathbb{N}\), \(m\in \mathbb N\), denote the \(m\)-ary Cantor set. Let \(x = x_1x_2\ldots\) and \(y= y_1y_2\ldots\) be points of \(\mathcal{C}^m\) and define a metric \(d\) on \(\mathcal{C}^m\) as follows. Fix an \(r\in (0,1)\) and set NEWLINE\[NEWLINE d(x, y) := r^k, NEWLINE\]NEWLINE where \(k+1\) is the least integer such that \(x_k\neq y_k\). The authors investigate the dimensions of the intersection \(D\) of a subset \(E\) of \(\mathcal{C}^m\) with the image of a subset \(F\) under random isometries (with respect to \(d\)). They obtain almost sure upper bounds for the Hausdorff and upper box-counting dimension of \(D\) and a lower bound for the essential supremum of the Hausdorff dimension.