Inhomogeneous self-similar sets and box dimensions

TL;DR

This paper studies the box dimensions of inhomogeneous self-similar sets, extending previous results for upper box dimensions and exploring the complex behavior of lower box dimensions through bounds and examples.

Contribution

It extends existing results on upper box dimensions and provides new insights and bounds on the more complex lower box dimension for inhomogeneous self-similar sets.

Findings

Upper box dimension computed under mild separation conditions

Lower box dimension exhibits irregular behavior compared to other dimensions

Examples demonstrate the complexity of lower box dimension

Abstract

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than the upper box dimension, Hausdorff dimension and packing dimension.