Dimensions of an overlapping generalization of Bara\'nski carpets

TL;DR

This paper calculates various fractal dimensions of a family of self-affine sets that generalize Barański carpets, allowing for overlaps and random translations, extending previous results on non-overlapping cases.

Contribution

It provides explicit formulas for Hausdorff, packing, and box-counting dimensions of overlapping Barański-like carpets, expanding the understanding of fractal dimensions under overlaps.

Findings

Dimensions coincide with non-overlapping case formulas outside small exceptional sets.

Box-counting and Hausdorff dimensions may differ in these fractals.

Results extend previous work on Bedford-McMullen carpets with overlaps.

Abstract

We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a Bara\'nski system and allow both vertical and horizontal random translations, while preserving the rows and columns structure. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulas will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and Hausdorff dimension do not necessarily agree. These results rely on M. Hochman's recent work on the dimensions of self-similar sets and measures, and can be seen as an extension of J. Fraser and P. Shmerkin results for Bedford-McMullen carpets with columns overlapping.