On the intersection of Cantor sets and products of random matrices
Nima Alibabaei
TL;DR
The paper advances the computation of the Hausdorff dimension for intersections of translated Cantor-type sets by linking it to the top Lyapunov exponent of random matrix products and introducing two complementary computational frameworks. The recurring method leverages affine co-invariance to yield closed-form dimension expressions when applicable, while the Neumann-series/kernel-expansion approach handles continuous stationary measures under the Neumann Admissibility Condition, delivering explicit lambda via a linear operator. A key result shows that for b≥7 with a single forbidden digit per Cantor component, the problem is kernel-expandable (or degenerate), enabling precise dimension calculations in several notable cases, including the middle-seventh Cantor intersection. The methods achieve polylogarithmic-time computation in favorable regimes, improving upon prior spectral-analytic truncation approaches and broadening the class of intersections for which exact dimensions can be efficiently determined. Overall, the work provides practical algorithms and rigorous criteria for computing dim_H in a broad combinatorial–analytic setting of translated Cantor sets.
Abstract
Kenyon and Peres (1991) showed that the Hausdorff dimension of intersections of randomly translated Cantor sets can be expressed in terms of the top Lyapunov exponent of a product of random matrices, and this exponent can be written as an integral with respect to stationary measures on the projective line. Although explicit computations are available when stationary measures are discrete, the continuous case has remained challenging. In this paper we introduce new combinatorial and analytic tools that allow us to compute the Lyapunov exponent, and hence the Hausdorff dimension, in a broad class of examples where stationary measures are continuous. As an application, we complete the dimension computation in the setting where a single digit is forbidden; for example, we determine the Hausdorff dimension of the intersection of the middle-seventh Cantor set with a random translate of itself.