Inhomogeneous self-similar sets with overlaps

TL;DR

This paper investigates the upper box dimension of inhomogeneous self-similar sets with overlaps, providing counterexamples to the expected dimension formula and establishing conditions under which it holds.

Contribution

It demonstrates that the standard dimension formula fails in the presence of overlaps and identifies the weak separation property as a condition for its validity.

Findings

Counterexamples show the formula does not hold with overlaps.

New upper bounds for the box dimension are established.

The expected formula holds under the weak separation property.

Abstract

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this `expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $SO(d)$ for $d\geq 3$. We also obtain new upper bounds for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the \emph{weak separation property} is satisfied, ie. the overlaps are controllable, then the `expected formula' does hold.