Hausdorff dimension of the set of singular pairs

• 成果类型: Article
• 署名作者: Cheung, Yitwah
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.1.4
• 发表日期: 2011
• 页码: 127-167
• 关键词: simultaneous diophantine approximations; divergent trajectories; homogeneous flows; n-tuples; foliations; SPACES

摘要

In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e(t), e(t), e(-2t)) on SL3R/SL(3)Z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A. N. Starkov. As a by-product of the analysis, we obtain a higher-dimensional generalization of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the technique used to compute Hausdorff dimension, we reprove a result of I. J. Good asserting that the Hausdorff dimension of the set of real numbers with divergent partial quotients is