Equidistribution from fractal measures

• 成果类型: Article
• 署名作者: Hochman, Michael; Shmerkin, Pablo
• 署名单位: University of Surrey
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0573-5
• 发表日期: 2015
• 页码: 427-479
• 关键词: hausdorff dimension; riesz products; normal numbers; invariant; set; projections; normality; RIGIDITY; entropy

摘要

We give a fractal-geometric condition for a measure on to be supported on points that are normal in base , i.e. such that equidistributes modulo 1. This condition is robust under coordinate changes, and it applies also when is a Pisot number rather than an integer. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.