Box-Counting Dimension in One-Dimensional Random Geometry of Multiplicative Cascades

We investigate the box-counting dimension of the image of a set $E\subset R$ under a random multiplicative cascade function f. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the context of random geometry, and for sufficiently regular sets, the same formula holds for the box-counting dimension. However, we show that this is far from true in general, and we compute explicitly a formula of a very different nature that gives the almost sure box-counting dimension of the random image f(E) when the set E comprises a convergent sequence. In particular, the box-counting dimension of f(E) depends more subtly on E than just on its dimensions. We also obtain lower and upper bounds for the box-counting dimension of the random images for general sets E.