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 $\tfrac12$.
-
[Ba77]
R. C. Baker, "Singular $n$-tuples and Hausdorff dimension," Math. Proc. Cambridge Philos. Soc., vol. 81, iss. 3, pp. 377-385, 1977.
@article {Ba77, MRKEY = {0435010},
AUTHOR = {Baker, R. C.},
TITLE = {Singular {$n$}-tuples and {H}ausdorff dimension},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {81},
YEAR = {1977},
NUMBER = {3},
PAGES = {377--385},
ISSN = {0305-0041},
MRCLASS = {10K15},
MRNUMBER = {0435010},
MRREVIEWER = {F. Schweiger},
DOI = {10.1017/S0305004100053457},
ZBLNUMBER = {0351.10041},
} -
[Ba92]
R. C. Baker, "Singular $n$-tuples and Hausdorff dimension. II," Math. Proc. Cambridge Philos. Soc., vol. 111, iss. 3, pp. 577-584, 1992.
@article {Ba92, MRKEY = {1151334},
AUTHOR = {Baker, R. C.},
TITLE = {Singular {$n$}-tuples and {H}ausdorff dimension. {II}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {111},
YEAR = {1992},
NUMBER = {3},
PAGES = {577--584},
ISSN = {0305-0041},
CODEN = {MPCPCO},
MRCLASS = {11J83 (28A78)},
MRNUMBER = {1151334},
MRREVIEWER = {F. Schweiger},
DOI = {10.1017/S030500410002171X},
ZBLNUMBER = {0755.11021},
} -
[Be]
A. S. Besicovitch, "Sets of fractional dimensions IV: On rational approximation to real numbers," J. London Math. Soc., vol. 9, pp. 126-131, 1934.
@article {Be, MRKEY = {0028380},
AUTHOR = {Besicovitch, A. S.},
TITLE = {Sets of fractional dimensions {IV}: {O}n rational approximation to real numbers},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {9},
YEAR = {1934},
PAGES = {126--131},
ZBLNUMBER={0009.05301},
DOI={10.1112/jlms/s1-9.2.126},
} -
[Ch03]
Y. Cheung, "Hausdorff dimension of the set of nonergodic directions," Ann. of Math., vol. 158, iss. 2, pp. 661-678, 2003.
@article {Ch03, MRKEY = {2018932},
AUTHOR = {Cheung, Yitwah},
TITLE = {Hausdorff dimension of the set of nonergodic directions},
NOTE = {with an appendix by M. Boshernitzan},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {158},
YEAR = {2003},
NUMBER = {2},
PAGES = {661--678},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {37D50 (11Z05 28A80 37A25 82C05)},
MRNUMBER = {2018932},
MRREVIEWER = {Serge L. Tabachnikov},
DOI = {10.4007/annals.2003.158.661},
ZBLNUMBER = {1037.37018},
} -
[Ch07]
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.
@article {Ch07, MRKEY = {2297087},
AUTHOR = {Cheung, Yitwah},
TITLE = {Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {27},
YEAR = {2007},
NUMBER = {1},
PAGES = {65--85},
ISSN = {0143-3857},
MRCLASS = {37C45 (37A17 37D40)},
MRNUMBER = {2297087},
MRREVIEWER = {Alexander Gorodnik},
DOI = {10.1017/S0143385706000678},
ZBLNUMBER = {1114.22013},
} -
[Da85]
S. G. Dani, "Divergent trajectories of flows on homogeneous spaces and Diophantine approximation," J. Reine Angew. Math., vol. 359, pp. 55-89, 1985.
@article {Da85, MRKEY = {794799},
AUTHOR = {Dani, S. G.},
TITLE = {Divergent trajectories of flows on homogeneous spaces and {D}iophantine approximation},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {359},
YEAR = {1985},
PAGES = {55--89},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {58F25 (22E40 58F11)},
MRNUMBER = {794799},
MRREVIEWER = {A. Morimoto},
DOI = {10.1515/crll.1985.359.55},
ZBLNUMBER = {0578.22012},
} -
[Da86]
S. G. Dani, "Bounded orbits of flows on homogeneous spaces," Comment. Math. Helv., vol. 61, iss. 4, pp. 636-660, 1986.
@article {Da86, MRKEY = {870710},
AUTHOR = {Dani, S. G.},
TITLE = {Bounded orbits of flows on homogeneous spaces},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {61},
YEAR = {1986},
NUMBER = {4},
PAGES = {636--660},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {22D40 (22E40 28D10 57S20 58F17)},
MRNUMBER = {88i:22011},
MRREVIEWER = {Arlan Ramsay},
DOI = {10.1007/BF02621936},
ZBLNUMBER = {0627.22013},
} -
[Fa03]
K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Second ed., Hoboken, NJ: John Wiley & Sons, 2003.
@book {Fa03, MRKEY = {2118797},
AUTHOR = {Falconer, Kenneth},
TITLE = {Fractal {{G}}eometry, {{M}}athematical {{F}}oundations and {{A}}pplications},
EDITION = {Second},
PUBLISHER = {John Wiley \& Sons},
ADDRESS = {Hoboken, NJ},
YEAR = {2003},
PAGES = {xxviii+337},
ISBN = {0-470-84861-8},
MRCLASS = {28-01 (00A69 11K55 28A75 28A78 28A80 37C45 37F10)},
MRNUMBER = {2006b:28001},
MRREVIEWER = {Esa A. J{ä}rvenp{ä}{ä}},
DOI = {10.1002/0470013850},
ZBLNUMBER = {1060.28005},
} -
[Ja] V. Jarn`ik, "Diophantischen approximationen und Hausdorffsches mass," Mat. Sbornik, vol. 36, pp. 371-382, 1929.
@article{Ja,
author={Jarnìk, V.},
TITLE={Diophantischen approximationen und Hausdorffsches mass},
JOURNAL={Mat. Sbornik},
VOLUME={36},
YEAR={1929},
PAGES={371--382},
JFMNUMBER={55.0719.01},
} -
[Kh26]
A. Khintchine, "Zur metrischen Theorie der diophantischen Approximationen," Math. Z., vol. 24, iss. 1, pp. 706-714, 1926.
@article {Kh26, MRKEY = {1544787},
AUTHOR = {Khintchine, A.},
TITLE = {Zur metrischen {T}heorie der diophantischen {A}pproximationen},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {24},
YEAR = {1926},
NUMBER = {1},
PAGES = {706--714},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {Contributed Item},
MRNUMBER = {1544787},
DOI = {10.1007/BF01216806},
} -
[KM96] D. Y. Kleinbock and G. A. Margulis, "Bounded orbits of nonquasiunipotent flows on homogeneous spaces," in Sina\u\i’s Moscow Seminar on Dynamical Systems, Providence, RI: Amer. Math. Soc., 1996, vol. 171, pp. 141-172.
@incollection {KM96, MRKEY = {1359098},
AUTHOR = {Kleinbock, D. Y. and Margulis, G. A.},
TITLE = {Bounded orbits of nonquasiunipotent flows on homogeneous spaces},
BOOKTITLE = {Sina\u\i's {M}oscow {S}eminar on {D}ynamical {S}ystems},
SERIES = {Amer. Math. Soc. Transl. Ser. 2},
VOLUME = {171},
PAGES = {141--172},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1996},
MRCLASS = {22E40 (58F15)},
MRNUMBER = {1359098},
MRREVIEWER = {Alexander Starkov},
ZBLNUMBER = {0843.22027},
} -
[KW1] 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.
@incollection {KW1, MRKEY = {2180240},
AUTHOR = {Kleinbock, Dmitry and Weiss, Barak},
TITLE = {Friendly measures, homogeneous flows and singular vectors},
BOOKTITLE = {Algebraic and {{T}}opological {{D}}ynamics},
SERIES = {Contemp. Math.},
VOLUME= {385},
PAGES = {281--292},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
MRCLASS = {11J83 (37A45 37D40)},
MRNUMBER = {218024},
MRREVIEWER = {Thomas Ward},
ZBLNUMBER = {1130.11040},
} -
[KW2] 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.
@article {KW2, MRKEY = {2366229},
AUTHOR = {Kleinbock, Dmitry and Weiss, Barak},
TITLE = {Dirichlet's theorem on {D}iophantine approximation and homogeneous flows},
JOURNAL = {J. Mod. Dyn.},
FJOURNAL = {Journal of Modern Dynamics},
VOLUME = {2},
YEAR = {2008},
NUMBER = {1},
PAGES = {43--62},
ISSN = {1930-5311},
MRCLASS = {11J83 (11J54 37A17 37A45)},
MRNUMBER = {2366229},
MRREVIEWER = {Thomas Ward},
ZBLNUMBER = {1143.11022},
} -
[KW04]
D. Kleinbock and B. Weiss, "Bounded geodesics in moduli space," Int. Math. Res. Not., vol. 2004, iss. 30, pp. 1551-1560, 2004.
@article {KW04, MRKEY = {2049831},
AUTHOR = {Kleinbock, Dmitry and Weiss, Barak},
TITLE = {Bounded geodesics in moduli space},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2004},
NUMBER = {30},
PAGES = {1551--1560},
ISSN = {1073-7928},
MRCLASS = {37D40 (53D25)},
MRNUMBER = {2049831},
MRREVIEWER = {William Abikoff},
DOI = {10.1155/S1073792804133412},
VOLUME = {2004},
ZBLNUMBER = {1075.37008},
} -
[JL1]
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.
@article {JL1, MRKEY = {662052},
AUTHOR = {Lagarias, J. C.},
TITLE = {Best simultaneous {D}iophantine approximations. {I}. {G}rowth rates of best approximation denominators},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {272},
YEAR = {1982},
NUMBER = {2},
PAGES = {545--554},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {10F10 (10F20)},
MRNUMBER = {0662052},
MRREVIEWER = {P. L. Cijsouw},
DOI = {10.2307/1998713},
} -
[JL2]
J. C. Lagarias, "Best simultaneous Diophantine approximations. II. Behavior of consecutive best approximations," Pacific J. Math., vol. 102, iss. 1, pp. 61-88, 1982.
@article {JL2, MRKEY = {682045},
AUTHOR = {Lagarias, J. C.},
TITLE = {Best simultaneous {D}iophantine approximations. {II}. {B}ehavior of consecutive best approximations},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {102},
YEAR = {1982},
NUMBER = {1},
PAGES = {61--88},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {10F10 (10F20)},
MRNUMBER = {682045},
MRREVIEWER = {P. L. Cijsouw},
ZBLNUMBER={0497.10025},
DOI={10.1090/S0002-9947-1982-0662052-7},
} -
[Ma92]
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.
@article {Ma92, MRKEY = {1167101},
AUTHOR = {Masur, Howard},
TITLE = {Hausdorff dimension of the set of nonergodic foliations of a quadratic differential},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {66},
YEAR = {1992},
NUMBER = {3},
PAGES = {387--442},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {30F30 (58F11 58F18)},
MRNUMBER = {1167101},
MRREVIEWER = {Masakazu Shiba},
DOI = {10.1215/S0012-7094-92-06613-0},
ZBLNUMBER = {0780.30032},
} -
[Mc87]
C. McMullen, "Area and Hausdorff dimension of Julia sets of entire functions," Trans. Amer. Math. Soc., vol. 300, iss. 1, pp. 329-342, 1987.
@article {Mc87, MRKEY = {871679},
AUTHOR = {McMullen, Curt},
TITLE = {Area and {H}ausdorff dimension of {J}ulia sets of entire functions},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {300},
YEAR = {1987},
NUMBER = {1},
PAGES = {329--342},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {30D05 (58F08 58F20)},
MRNUMBER = {88a:30057},
MRREVIEWER = {I. N. Baker},
DOI = {10.2307/2000602},
ZBLNUMBER = {0618.30027},
} -
[Mo04]
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.
@incollection {Mo04, MRKEY = {2306143},
AUTHOR = {Moshchevitin, Nikolai G.},
TITLE = {Best {D}iophantine approximations: the phenomenon of degenerate dimension},
BOOKTITLE = {Surveys in {{G}}eometry and {{N}}umber {{T}}heory: {{R}}eports on {{C}}ontemporary {R}ussian {{M}}athematics},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {338},
PAGES = {158--182},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2007},
MRCLASS = {11J13 (11K60)},
MRNUMBER = {2306143},
MRREVIEWER = {Lenny Fukshansky},
ZBLNUMBER = {1151.11031},
DOI={10.1017/CBO9780511721472.006},
} -
[MS91]
H. Masur and J. Smillie, "Hausdorff dimension of sets of nonergodic measured foliations," Ann. of Math., vol. 134, iss. 3, pp. 455-543, 1991.
@article {MS91, MRKEY = {1135877},
AUTHOR = {Masur, Howard and Smillie, John},
TITLE = {Hausdorff dimension of sets of nonergodic measured foliations},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {134},
YEAR = {1991},
NUMBER = {3},
PAGES = {455--543},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {58F18 (28A78 30F30 57M50 57R30 58F11)},
MRNUMBER = {1135877},
MRREVIEWER = {Darryl McCullough},
DOI = {10.2307/2944356},
ZBLNUMBER = {0774.58024},
} -
[Re] H. Reeve, On the Hausdorff dimension of sets of uniformly approximable numbers.
@misc{Re,
author={Reeve, H.},
TITLE={{\rm {O}n the Hausdorff dimension of sets of uniformly approximable numbers}},
NOTE={preprint},
} -
[Ry90]
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.
@article {Ry90, MRKEY = {1027791},
AUTHOR = {Rynne, Bryan P.},
TITLE = {A lower bound for the {H}ausdorff dimension of sets of singular {$n$}-tuples},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {107},
YEAR = {1990},
NUMBER = {2},
PAGES = {387--394},
ISSN = {0305-0041},
CODEN = {MPCPCO},
MRCLASS = {11J13 (28A78)},
MRNUMBER = {1027791},
MRREVIEWER = {F. Schweiger},
DOI = {10.1017/S0305004100068651},
ZBLNUMBER = {0707.11056},
} -
[Sc66]
W. M. Schmidt, "On badly approximable numbers and certain games," Trans. Amer. Math. Soc., vol. 123, pp. 178-199, 1966.
@article {Sc66, MRKEY = {0195595},
AUTHOR = {Schmidt, Wolfgang M.},
TITLE = {On badly approximable numbers and certain games},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {123},
YEAR = {1966},
PAGES = {178--199},
ISSN = {0002-9947},
MRCLASS = {90.70},
MRNUMBER = {0195595},
MRREVIEWER = {T. M. Cover},
DOI = {10.2307/1994619},
ZBLNUMBER = {0232.10029},
} -
[St] A. N. Starkov, Dynamical Systems on Momogeneous Spaces, Providence, RI: Amer. Math. Soc., 2000, vol. 190.
@book {St, MRKEY = {1746847},
AUTHOR = {Starkov, Alexander N.},
TITLE = {Dynamical {{S}}ystems on {{M}}omogeneous {{S}}paces},
SERIES = {Transl. Math. Monogr.},
VOLUME= {190},
NOTE = {translated from the 1999 Russian original by the author},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {xvi+243},
ISBN = {0-8218-1389-7},
MRCLASS = {37A25 (11E04 11H55 22E40 37C85)},
MRNUMBER = {1746847},
MRREVIEWER = {S. G. Dani},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {1143.37300},
} -
[W1]
B. Weiss, "Divergent trajectories on noncompact parameter spaces," Geom. Funct. Anal., vol. 14, iss. 1, pp. 94-149, 2004.
@article {W1, MRKEY = {2053601},
AUTHOR = {Weiss, Barak},
TITLE = {Divergent trajectories on noncompact parameter spaces},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {14},
YEAR = {2004},
NUMBER = {1},
PAGES = {94--149},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {22E40 (37A15 37B99 37C80 37D40)},
MRNUMBER = {2053601},
MRREVIEWER = {Mikhail S. Kulikov},
DOI = {10.1007/s00039-004-0453-z},
ZBLNUMBER = {1074.37004},
} -
[W2]
B. Weiss, "Divergent trajectories and $\Bbb Q$-rank," Israel J. Math., vol. 152, pp. 221-227, 2006.
@article {W2, MRKEY = {2214461},
AUTHOR = {Weiss, Barak},
TITLE = {Divergent trajectories and {$\Bbb Q$}-rank},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {152},
YEAR = {2006},
PAGES = {221--227},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {22E40},
MRNUMBER = {2214461},
MRREVIEWER = {Lizhen Ji},
DOI = {10.1007/BF02771984},
ZBLNUMBER = {1126.22007},
} -
[Ya87] 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.
@article {Ya87, MRKEY = {912957},
AUTHOR = {Yavid, K. Yu.},
TITLE = {An estimate for the {H}ausdorff dimension of a set of singular vectors},
JOURNAL = {Dokl. Akad. Nauk BSSR},
FJOURNAL = {Doklady Akademii Nauk BSSR},
VOLUME = {31},
YEAR = {1987},
NUMBER = {9},
PAGES = {777--780, 859},
ISSN = {0002-354X},
MRCLASS = {11K55 (11H99 11J99)},
MRNUMBER = {912957},
MRREVIEWER = {Yao Chen Zhu},
ZBLNUMBER = {0634.10044},
}