Annals of Mathematics· 2014Q1
On self-similar sets with overlaps and inverse theorems for entropy
- 278citations
- Q1SCImago
- 2014year
Short summary
A new inverse theorem on entropy growth under convolution reveals that dimension drops in self-similar sets imply superexponentially close cylinders, confirming exact overlaps for algebraic coefficients.
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Key points
- Dimension drops in self-similar sets imply superexponentially close cylinders at small scales.
- Exact overlaps are confirmed for self-similar sets with algebraic coefficients.
- An inverse theorem on entropy growth under convolution is a key tool.
- Applications include Furstenberg's conjecture and Bernoulli convolutions.
AI-generated from the title and abstract; the full text is not read.
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 min{1, s}, 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 coefficients. 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.
The authors' abstract, as published at the source. Annals of Mathematics, 2014 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics