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We estimate the Hausdorff measure and dimension of Cantor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the dimension of the associated Cantor set.It is well-known that not every Cantor set on the line is an s-set for some 0 ≤ s ≤ 1. However, if the sequence associated to the Cantor set C is nonincreasing, we show that C is an h-set for some continuous, concave dimension function h. We construct the function h from the sequence associated to the set C.

Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0, 1] such that the lengths of their complementary open intervals are given by the terms of a, the so-called complementary sets. In this paper we determine the almost sure value of the Φ-dimensions of these sets given a natural model of randomness. The Φ-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of Φ, with one size behaving like the Assouad dimension and the other, like the quasi-Assouad dimension.

We analyze a multi-sector growth model subject to random shocks affecting the two sector-specific production functions twofold: the evolution of both productivity and factor shares is the result of such exogenous shocks. We determine the optimal dynamics via Euler-Lagrange equations, and show how these dynamics can be described in terms of an iterated function system with probability. We also provide conditions that imply the singularity of the invariant measure associated with the fractal attractor. Numerical examples show how specific parameter configurations might generate distorted copies of the Barnsley's fern attractor.