## Abstract

We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of minf1; sg, where s is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coeffcients. As applications we prove Furstenberg's conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2^{-n} has only a small amount of growth under convolution.

Original language | American English |
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Pages (from-to) | 773-822 |

Number of pages | 50 |

Journal | Annals of Mathematics |

Volume | 180 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |