Calculating box dimension with the method of types

Abstract

This paper presents a general procedure based on using the method of types to calculate the box dimension of sets. The approach unifies and simplifies multiple box counting arguments. In particular, we use it to generalize the formula for the box dimension of self-affine carpets of Gatzouras-Lalley and of Barański type to their higher dimensional sponge analogues. In addition to a closed form, we also obtain a variational formula which resembles the Ledrappier-Young formula for Hausdorff dimension.

arXiv comment14 pages, 1 figure

References in corpus (6)

• Ledrappier-Young formula and exact dimensionality of self-affine measures
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Cited by in corpus (1)

• Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence