Hausdorff dimension of the set of singular pairs

Abstract

In this paper we show that the Hausdorff dimension of the set of singular pairs is $\tfrac{4}{3}$. We also show that the action of $\mathrm{diag}(e^t,e^t,e^{-2t})$ on $\mathrm{SL}_3 \mathbb{R}/\mathrm{SL}_3\mathbb{Z}$ 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 $\tfrac12$.