On dense intermingling of exact overlaps and the open set condition

Abstract

We prove that certain families of homogenous affine iterated function systems in $\mathbb{R}^d$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan-Pollicott-Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb{R}^d$ with an adaptation of a classic argument of Erd\H{o}s on the singularity of certain Bernoulli convolutions. Our result encompasses a one-dimensional example due to Kenyon which arises as a special case.