Hausdorff dimension of the set of singular pairs

By Y. Cheung. 2007

95 citations recorded by the source corpus.

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 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 1 2 .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d45dfbb9219f046d6b6710366be279b81238bf2

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Singular systems of linear forms and non-escape of mass in the space of lattices (2014, background): “zed by Dani. Thus the set of points with divergent orbits has a quite complicated structure. In particular, computing its Hausdor dimension is a dicult problem. When m+n= 3 it was shown by Y. Cheung [Che11] that the Hausdor dimension of the set of singular pairs is 4=3, hence the set of points in X ”
A Poincar\'e section for horocycle flow on the space of lattices (2012, result): “In this way it is similar to work of the second author [14], where a piecewise linear description of diagonal flows on SL(3,R)/SL(3,Z) was studied.”