Hausdorff dimension of the set of singular pairs

2007/09/28 by Yitwah Cheung, Cheung, Yitwah · 2 citations
Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #advanced mathematical theories #math.DS #math.NT #msc:11J70 #msc:11K40 #msc:22E40 #msc:37A17

paper · pdf · doi:10.48550/arxiv.0709.4534

40 pages, revised proofs, added explanations

arxiv created 2008/10/22 · arxiv updated 2009/12/01

Abstract

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 SL(3,R)/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 generalisation of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the techniques used to compute Hausdorff dimension, we show that the set of real numbers with divergent partial quotients has Hausdorff dimension 1/2.

Cited by

• Slowly Divergent Trajectories for Weighted Singular Vectors in R²
• Dichotomy for the Hausdorff dimension of nonergodic directions on translation surfaces

Related

• On divergent on average trajectories for higher rank actions
• A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions
• Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space
• On the Hausdorff dimension of certain sets arising in Number Theory
• Divergent trajectories on products of homogeneous spaces