### 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-α projections of the natural measure of the Sierpiński carpet. We present more general one-parameter families of self-similar measures vα, such that the set of parameters α for which vα is singular is a dense G _{δ} set but this “exceptional” set of parameters of singularity has zero Hausdorff dimension.

Original language | English |
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Pages (from-to) | 1971-1986 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 2019 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*147*(5), 1971-1986. https://doi.org/10.1090/proc/14042