On self-similar sets with overlaps and inverse theorems for entropy

• 成果类型: Article
• 署名作者: Hochman, Michael
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2014.180.2.7
• 发表日期: 2014
• 页码: 773-822
• 关键词: hausdorff dimension; sum-product; smoothness; expansions; systems; series; FAMILY

摘要

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{l, s}, where 8 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 has only a small amount of growth under convolution.