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

Abstract We investigate the box-counting dimension of the image of a set $$E \subset \mathbb {R}$$ E ⊂ 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 .