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Mat. Sb., 2003, Volume 194, Number 2, Pages 129–144 (Mi msb717)  

This article is cited in 25 scientific papers (total in 25 papers)

Inhomogeneous Diophantine approximation and angular recurrence for polygonal billiards

S. Troubetzkoy, J. Schmeling

Institut de Mathématiques de Luminy

Abstract: For a fixed rotation number we compute the Hausdorff dimension of the set of well approximable numbers. We use this result and an inhomogeneous version of Jarnik's theorem to demonstrate strong recurrence properties of the billiard flow in certain polygons.

DOI: https://doi.org/10.4213/sm717

Full text: PDF file (331 kB)
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English version:
Sbornik: Mathematics, 2003, 194:2, 295–309

Bibliographic databases:

UDC: 517.987
MSC: 37D50, 11J83
Received: 17.05.2001 and 21.06.2002

Citation: S. Troubetzkoy, J. Schmeling, “Inhomogeneous Diophantine approximation and angular recurrence for polygonal billiards”, Mat. Sb., 194:2 (2003), 129–144; Sb. Math., 194:2 (2003), 295–309

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Monteil T., “On the finite blocking property”, Ann. Inst. Fourier (Grenoble), 55:4 (2005), 1195–1217  crossref  mathscinet  zmath  isi
    2. Troubetzkoy S., “Periodic billiard orbits in right triangles”, Ann. Inst. Fourier (Grenoble), 55:1 (2005), 29–46  crossref  mathscinet  zmath  isi
    3. Ya. Bugeaud, M. Laurent, “On exponents of homogeneous and inhomogeneous Diophantine approximation”, Mosc. Math. J., 5:4 (2005), 747–766  mathnet  crossref  mathscinet  zmath
    4. Dodson M.M., Kristensen S., “Khintchine's theorem and transference principle for star bodies”, Int. J. Number Theory, 2:3 (2006), 431–453  crossref  mathscinet  zmath  isi
    5. Bugeaud Y., Chevallier N., “On simultaneous inhomogeneous Diophantine approximation”, Acta Arith., 123:2 (2006), 97–123  crossref  mathscinet  zmath  adsnasa  isi
    6. Fan Ai-Hua, Wu Jun, “A note on inhomogeneous Diophantine approximation with a general error function”, Glasg. Math. J., 48 (2006), 187–191  crossref  mathscinet  zmath  isi  elib
    7. Kim Dong Han, “The shrinking target property of irrational rotations”, Nonlinearity, 20:7 (2007), 1637–1643  crossref  mathscinet  zmath  adsnasa  isi
    8. M. Laurent, “On inhomogeneous Diophantine approximation and Hausdorff dimension”, J. Math. Sci., 180:5 (2012), 592–598  mathnet  crossref  mathscinet  elib
    9. Bugeaud Ya., Harrap S., Kristensen S., Velani S., “On Shrinking Targets for Z(M) Actions on Tori”, Mathematika, 56:2 (2010), 193–202  crossref  mathscinet  zmath  isi
    10. LINGMIN LIAO, STÉPHANE SEURET, “Diophantine approximation by orbits of expanding Markov maps”, Ergod. Th. Dynam. Sys, 2012, 1  crossref  mathscinet  zmath  isi
    11. Troubetzkoy S., “Recurrence in Generic Staircases”, Discrete and Continuous Dynamical Systems, 32:3 (2012), 1047–1053  crossref  mathscinet  zmath  isi
    12. LuMing Shen, BaoWei Wang, “Shrinking target problems for beta-dynamical system”, Sci. China Math, 2012  crossref  mathscinet  isi
    13. D.H.an Kim, Bo Tan, Baowei Wang, Jian Xu, “Kurzweil type metrical Diophantine properties in the field of formal Laurent series”, Journal of Mathematical Analysis and Applications, 2013  crossref  mathscinet  isi
    14. Bing Li, Tomas Persson, Baowei Wang, Jun Wu, “Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions”, Math. Z, 2013  crossref  mathscinet  isi
    15. Fan A.-H., Schmeling J., Troubetzkoy S., “A Multifractal MASS Transference Principle for Gibbs Measures with Applications to Dynamical Diophantine Approximation”, Proc. London Math. Soc., 107:5 (2013), 1173–1219  crossref  mathscinet  zmath  isi
    16. Liao L., Rams M., “Inhomogeneous Diophantine Approximation with General Error Functions”, Acta Arith., 160:1 (2013), 25–35  crossref  mathscinet  zmath  isi
    17. Li B., Wang B.-W., Wu J., Xu J., “The Shrinking Target Problem in the Dynamical System of Continued Fractions”, Proc. London Math. Soc., 108:1 (2014), 159–186  crossref  mathscinet  zmath  isi
    18. Ma Ch., Wang Sh., “Dynamical Diophantine Approximation of Beta Expansions of Formal Laurent Series”, Finite Fields their Appl., 34 (2015), 176–191  crossref  mathscinet  zmath  isi
    19. Bugeaud Y. Durand A., “Metric Diophantine approximation on the middle-third Cantor set”, J. Eur. Math. Soc., 18:6 (2016), 1233–1272  crossref  mathscinet  zmath  isi  scopus
    20. Beresnevich V. Ramirez F. Velani S., “Metric Diophantine Approximation: Aspects of Recent Work”, Dynamics and Analytic Number Theory, London Mathematical Society Lecture Note Series, ed. Badziahin D. Gorodnik A. Peyerimhoff N., Cambridge Univ Press, 2016, 1–95  mathscinet  zmath  isi
    21. Wang B., Wu J., “A Survey on the Dimension Theory in Dynamical Diophantine Approximation”, Recent Developments in Fractals and Related Fields, Trends in Mathematics, eds. Barral J., Seuret S., Birkhauser Verlag Ag, 2017, 261–294  crossref  mathscinet  zmath  isi  scopus
    22. Kim D.H., Rams M., Wang B., “Hausdorff Dimension of the Set Approximated By Irrational Rotations”, Mathematika, 64:1 (2018), 267–283  crossref  mathscinet  zmath  isi
    23. Barany B., Rams M., “Shrinking Targets on Bedford-Mcmullen Carpets”, Proc. London Math. Soc., 117:5 (2018), 951–995  crossref  mathscinet  zmath  isi  scopus
    24. Ekstrom F., Persson T., “Hausdorff Dimension of Random Limsup Sets”, J. Lond. Math. Soc.-Second Ser., 98:3 (2018), 661–686  crossref  mathscinet  zmath  isi  scopus
    25. Kim D.H., Liao L., “Dirichlet Uniformly Well-Approximated Numbers”, Int. Math. Res. Notices, 2019:24 (2019), 7691–7732  crossref  mathscinet  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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