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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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##### References:
 [1] R. C. Baker, ”Singular $$n$$-tuples and Hausdorff dimension,” Math. Proc. Cambridge Philos. Soc., vol. 81, iss. 3, pp. 377-385, 1977. · Zbl 0351.10041 · doi:10.1017/S0305004100053457 [2] R. C. Baker, ”Singular $$n$$-tuples and Hausdorff dimension. II,” Math. Proc. Cambridge Philos. Soc., vol. 111, iss. 3, pp. 577-584, 1992. · Zbl 0755.11021 · doi:10.1017/S0305004100075654 [3] A. S. Besicovitch, ”Sets of fractional dimensions IV: On rational approximation to real numbers,” J. London Math. Soc., vol. 9, pp. 126-131, 1934. · Zbl 0009.05301 · doi:10.1112/jlms/s1-9.2.126 [4] Y. Cheung, ”Hausdorff dimension of the set of nonergodic directions,” Ann. of Math., vol. 158, iss. 2, pp. 661-678, 2003. · Zbl 1037.37018 · doi:10.4007/annals.2003.158.661 · euclid:annm/1069786257 · arxiv:math/0404433 [5] Y. Cheung, ”Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space,” Ergodic Theory Dynam. Systems, vol. 27, iss. 1, pp. 65-85, 2007. · Zbl 1114.22013 · doi:10.1017/S0143385706000678 · arxiv:math/0608002 [6] S. G. Dani, ”Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,” J. Reine Angew. Math., vol. 359, pp. 55-89, 1985. · Zbl 0578.22012 · doi:10.1515/crll.1985.359.55 · crelle:GDZPPN002202522 · eudml:152734 [7] S. G. Dani, ”Bounded orbits of flows on homogeneous spaces,” Comment. Math. Helv., vol. 61, iss. 4, pp. 636-660, 1986. · Zbl 0627.22013 · doi:10.1007/BF02621936 · eudml:140072 [8] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Second ed., Hoboken, NJ: John Wiley & Sons, 2003. · Zbl 1060.28005 · doi:10.1002/0470013850 [9] V. Jarn‘ik, ”Diophantischen approximationen und Hausdorffsches mass,” Mat. Sbornik, vol. 36, pp. 371-382, 1929. · JFM 55.0719.01 [10] A. Khintchine, ”Zur metrischen Theorie der diophantischen Approximationen,” Math. Z., vol. 24, iss. 1, pp. 706-714, 1926. · JFM 52.0183.02 · doi:10.1007/BF01216806 [11] D. Y. Kleinbock and G. A. Margulis, ”Bounded orbits of nonquasiunipotent flows on homogeneous spaces,” in Sinaĭ’s Moscow Seminar on Dynamical Systems, Providence, RI: Amer. Math. Soc., 1996, vol. 171, pp. 141-172. · Zbl 0843.22027 [12] D. Kleinbock and B. Weiss, ”Friendly measures, homogeneous flows and singular vectors,” in Algebraic and Topological Dynamics, Providence, RI: Amer. Math. Soc., 2005, vol. 385, pp. 281-292. · Zbl 1130.11040 · arxiv:math/0506513 [13] D. Kleinbock and B. Weiss, ”Dirichlet’s theorem on Diophantine approximation and homogeneous flows,” J. Mod. Dyn., vol. 2, iss. 1, pp. 43-62, 2008. · Zbl 1143.11022 · arxiv:math/0612171 [14] D. Kleinbock and B. Weiss, ”Bounded geodesics in moduli space,” Int. Math. Res. Not., vol. 2004, iss. 30, pp. 1551-1560, 2004. · Zbl 1075.37008 · doi:10.1155/S1073792804133412 · arxiv:math/0304029 [15] J. C. Lagarias, ”Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators,” Trans. Amer. Math. Soc., vol. 272, iss. 2, pp. 545-554, 1982. · Zbl 0495.10021 · doi:10.2307/1998713 [16] J. C. Lagarias, ”Best simultaneous Diophantine approximations. II. Behavior of consecutive best approximations,” Pacific J. Math., vol. 102, iss. 1, pp. 61-88, 1982. · Zbl 0497.10025 · doi:10.2140/pjm.1982.102.61 [17] H. Masur, ”Hausdorff dimension of the set of nonergodic foliations of a quadratic differential,” Duke Math. J., vol. 66, iss. 3, pp. 387-442, 1992. · Zbl 0780.30032 · doi:10.1215/S0012-7094-92-06613-0 [18] C. McMullen, ”Area and Hausdorff dimension of Julia sets of entire functions,” Trans. Amer. Math. Soc., vol. 300, iss. 1, pp. 329-342, 1987. · Zbl 0618.30027 · doi:10.2307/2000602 [19] N. G. Moshchevitin, ”Best Diophantine approximations: the phenomenon of degenerate dimension,” in Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics, Cambridge: Cambridge Univ. Press, 2007, vol. 338, pp. 158-182. · Zbl 1151.11031 · doi:10.1017/CBO9780511721472.006 · arxiv:math/0411293 [20] H. Masur and J. Smillie, ”Hausdorff dimension of sets of nonergodic measured foliations,” Ann. of Math., vol. 134, iss. 3, pp. 455-543, 1991. · Zbl 0774.58024 · doi:10.2307/2944356 [21] H. Reeve, On the Hausdorff dimension of sets of uniformly approximable numbers. [22] B. P. Rynne, ”A lower bound for the Hausdorff dimension of sets of singular $$n$$-tuples,” Math. Proc. Cambridge Philos. Soc., vol. 107, iss. 2, pp. 387-394, 1990. · Zbl 0707.11056 · doi:10.1017/S0305004100068651 [23] W. M. Schmidt, ”On badly approximable numbers and certain games,” Trans. Amer. Math. Soc., vol. 123, pp. 178-199, 1966. · Zbl 0232.10029 · doi:10.2307/1994619 [24] A. N. Starkov, Dynamical Systems on Momogeneous Spaces, Providence, RI: Amer. Math. Soc., 2000, vol. 190. · Zbl 1143.37300 [25] B. Weiss, ”Divergent trajectories on noncompact parameter spaces,” Geom. Funct. Anal., vol. 14, iss. 1, pp. 94-149, 2004. · Zbl 1074.37004 · doi:10.1007/s00039-004-0453-z [26] B. Weiss, ”Divergent trajectories and $$\mathbb Q$$-rank,” Israel J. Math., vol. 152, pp. 221-227, 2006. · Zbl 1126.22007 · doi:10.1007/BF02771984 [27] Y. K. Yavid, ”An estimate for the Hausdorff dimension of a set of singular vectors,” Dokl. Akad. Nauk BSSR, vol. 31, iss. 9, pp. 777-780, 859, 1987. · Zbl 0634.10044
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