The box-counting dimension in one-dimensional random geometry of multiplicative cascades

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 talk 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 bounds on the dimension in the random metric. We will also show that there is no hope for an exact formula through a surprisingly simple family of countable sets (sequences accumulating at 0). These examples show that the dimension with the random metric depends on more structural information and that there cannot be a KPZ equation for the box-counting dimension. 

(Joint with Kenneth Falconer)