Box-counting dimension in one-dimensional random geometry of multiplicative cascades

Abstract

A result of Benjamini and Schramm shows that the Hausdorff dimension of sets in one-dimensional random geometry given by multiplicative cascades satisfies an elegant formula dependent only on the random variable and the dimension of the set in Euclidean geometry. In this article we show that this holds for the box-counting dimension when the set is sufficiently regular. This formula, however, is not valid in general and we provide general bounds on the box-counting dimension in the random metric. We explicitly compute the box-counting dimension for a large family of countable sets that accumulate at a single point which shows that the Benjamini-Schramm type formula cannot hold in general. This shows that the situation for the box-counting dimension is more subtle and knowledge of the structure is needed. We illustrate our results by providing examples including a pair of sets with the same box-counting dimension but different dimensions in the random metric.