Dimensions of an overlapping generalization of Bara\'nski carpets

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.