A proof of Furstenberg's conjecture on the intersections of xp- and xq-invariant sets

• 成果类型: Article
• 署名作者: Wu, Meng
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2019.189.3.2
• 发表日期: 2019
• 页码: 707-751
• 关键词: affine embeddings; cantor sets; projections; dimension; sections

摘要

We prove the following conjecture of Furstenberg (1969): if A, B subset of [0,1] are closed and invariant under xp mod 1 and xq mod 1, respectively, and if log p/log q is not an element of Q, then for all real numbers u and v, dim(H) (uA+ v) boolean AND B <= max {0, dim(H) A + dim(H) B - 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on R. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.