On the dimensions of a family of overlapping self-affine carpets

TL;DR

This paper investigates the dimensions of a family of self-affine sets related to Bedford-McMullen carpets, analyzing how random translations affect their Hausdorff, packing, and box dimensions, extending Hochman's recent work.

Contribution

It introduces a method to compute dimensions of randomized Bedford-McMullen-like sets, including overlapping cases, extending existing dimension theory for self-affine sets.

Findings

Dimensions are computed outside a small set of exceptional translations.

Explicit translations with overlaps still allow dimension calculations.

Results extend Hochman's work on self-similar sets to self-affine carpets.

Abstract

We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford-McMullen set up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, M. Hochman's recent work on the dimensions of self-similar sets and measures.