Singularity versus exact overlaps for self-similar measures

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 Sierpiński 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.