Superbranching processes and projections of random Cantor sets

We study sequences (X₀, X₁, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that , for m, n0, where the X_n(m, i) are distributed as X_n and have certain properties of independence. We prove that, under appropriate conditions, X_n^1/n almost surely and in L¹, where =sup E(X_n)^1/n. Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1]^d. We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant