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Hausdorff dimension of the set of singular pairs. (English) Zbl 1241.11075
A vector \({\mathbf x} \in {\mathbb R}^d\) is said to be singular if for any \(\delta > 0\), there is a \(T_0 > 0\) such that for any \(T > T_0\), \[ | q {\mathbf x} - {\mathbf p}| < {{\delta}\over{T^{1/d}}}, \quad 0 < q < T, \] has a solution \((q, {\mathbf p}) \in {\mathbb Z}^{d+1}\). The set of singular vectors is denoted \(\text{Sing}(d)\). In the present paper, it is shown that the Hausdorff dimension of the set \(\text{Sing}(2)\) is equal to \(4/3\). This has implications in homogeneous dynamics, where it answers a question of [A. N. Starkov, Dynamical systems on homogeneous spaces. Translations of Mathematical Monographs 190. Providence, RI: American Mathematical Society (2000; Zbl 1143.37300)] on the existence of slowly growing trajectories of a certain one-parameter action on \(\text{SL}_3({\mathbb R})/\text{SL}_3({\mathbb Z})\).
An impressing feature (aside from the above main results) of this well-written paper is the extension of some fundamental inequalities from the theory of continued fractions to the multidimensional setting in any dimension.
As an additional by-product, the machinery used to get upper and lower bounds for the Hausdorff dimension for the set \(\text{Sing}(2)\) is likely to have wider applicability. An example of this is already found in the paper, where the author uses the machinery to re-prove the classical result of I. J. Good [Proc. Camb. Philos. Soc. 37, 199–228 (1941; JFM 67.0988.03)] that the set of reals whose sequence of partial quotients tend to infinity is of Hausdorff dimension \(1/2\).

MSC:
11J70 Continued fractions and generalizations
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
22E40 Discrete subgroups of Lie groups
37A17 Homogeneous flows
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