On dense intermingling of exact overlaps and the open set condition

TL;DR

This paper shows that for certain affine iterated function systems, the open set condition and exact overlaps occur densely in parameter space, revealing complexities in self-affine set dimensions.

Contribution

It demonstrates the dense occurrence of both the open set condition and exact overlaps in families of self-affine systems, challenging previous assumptions about their typical behavior.

Findings

Open set condition and exact overlaps are dense in parameter space.

Dense exceptional sets affect the dimension theorems of Falconer and Jordan-Pollicott-Simon.

Includes a one-dimensional example by Kenyon as a special case.

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.