Hausdorff dimension of the set of singular pairs

TL;DR

This paper determines the Hausdorff dimension of the set of singular pairs as 4/3, explores divergent trajectories in a specific dynamical system, and generalizes inequalities related to continued fractions.

Contribution

It establishes the Hausdorff dimension of singular pairs, answers a question on divergent trajectories, and extends inequalities for continued fractions to higher dimensions.

Findings

Hausdorff dimension of singular pairs is 4/3

Divergent trajectories can exit at arbitrarily slow rates

Set of real numbers with divergent partial quotients has Hausdorff dimension 1/2

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.