Singularity versus exact overlaps for self-similar measures

TL;DR

This paper constructs families of self-similar measures with dimensions exceeding one, demonstrating that singularity can occur without exact overlaps, and characterizes the size of the singular set.

Contribution

It introduces new families of self-similar measures with high dimension where singularity is not due to overlaps, and analyzes the measure of parameters leading to singularity.

Findings

Existence of measure families with dimension > 1 and no exact overlaps causing singularity.

The set of parameters with singular measures is dense but has zero Hausdorff dimension.

Singularity can occur in a measure without the presence of exact overlaps.

Abstract

In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than $1$ for all parameters and - the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-$\alpha$ projections of the natural measure of the Sierpi\'nski carpet. We present more general one-parameter families of self-similar measures $\nu_\alpha$, such that the set of parameters $\alpha$ for which $\nu_\alpha$ is singular is a dense $G_\delta$ set but this "exceptional" set of parameters of singularity has zero Hausdorff dimension.