Superbranching processes and projections of random Cantor sets

Abstract

We study sequences (X 0, X 1, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that $$X_{m + n} \geqq \sum\limits_{i = 1}^{X_m } {X_n (m,i)}$$ , for m, n≧0, where the X n (m, i) are distributed as X n and have certain properties of independence. We prove that, under appropriate conditions, X 1/ →λ almost surely and in L 1, where λ=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.