Glasnik Matematicki, Vol. 50, No. 2 (2015), 349-361.

ON APPROXIMATION CONSTANTS FOR LIOUVILLE NUMBERS

Johannes Schleischitz

Institute of Mathematics, Univ. Nat. Res. Life Sci., Gregor-Mendel-Strasse 33, Vienna, 1180, Austria
e-mail: johannes.schleischitz@boku.ac.at


Abstract.   We investigate some Diophantine approximation constants related to the simultaneous approximation of (ζ,ζ2, ...,ζk) for Liouville numbers ζ. For a certain class of Liouville numbers including the famous representative n≥ 1 10-n! and numbers in the Cantor set, we explicitly determine all approximation constants simultaneously for all k≥ 1.

2010 Mathematics Subject Classification.   11J13, 11J25, 11J82.

Key words and phrases.   Geometry of numbers, successive minima, Liouville numbers, Diophantine approximation.


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DOI: 10.3336/gm.50.2.06


References:

  1. V. Beresnevich, D. Dickinson and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846.
    MathSciNet     CrossRef

  2. V. Beresnevich, D. Dickinson and S. L. Velani, Diophantine approximation on planar curves and the distribution of rational points with an appendix by R.C. Vaughan, Sums of two squares near perfect squares, Ann. Math. (2) 166 (2007), 367-426.
    MathSciNet     CrossRef

  3. Y. Bugeaud, On simultaneous rational approximation to a real numbers and its integral powers, Ann. Inst. Fourier (Grenoble) 60 (2010), 2165-2182.
    MathSciNet     CrossRef

  4. Y. Bugeaud and M. Laurent, Exponents of Diophantine Approximation and Sturmian Continued Fractions, Ann. Inst. Fourier (Grenoble) 55 (2005), 773-804.
    MathSciNet     CrossRef

  5. Y. Bugeaud, Nombres de Liouville et nombres normaux, C. R. Math. Acad. Sci. Paris, 335 (2002), 117-120.
    MathSciNet     CrossRef

  6. V. Jarník, Contribution á la théorie des approximations diophantiennes linéaires et homogénes, Czech Math. J. 4(79) (1954), 330-353.
    MathSciNet    

  7. A. Y. Khintchine, Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706-714
    MathSciNet     CrossRef

  8. D. Kleinbock, Extremal subspaces and their submanifolds, Geom. Funct. Anal. 13 (2003), 437-466.
    MathSciNet     CrossRef

  9. M. Laurent, Simultaneous rational approximation to successive powers of a real number, Indag. Math. (N.S.) 14 (2003), 45-53.
    MathSciNet     CrossRef

  10. K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. I, J. Reine Angew. Math. 166 (1932), 118-136.
    MathSciNet     CrossRef

  11. K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. II, J. Reine Angew. Math. 166 (1932), 137-150.
    MathSciNet     CrossRef

  12. H. Minkowski, Geometrie der Zahlen, Bibliotheca Mathematica Teubneriana, New York-London, 1968.
    MathSciNet    

  13. K.F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20.
    MathSciNet     CrossRef

  14. D. Roy, Diophantine approximation in small degree, Number theory, E.Z. Goren and H. Kisilevsky Eds, CRM Proceedings and Lecture Notes, 2004, p. 269-285.
    MathSciNet    

  15. D. Roy, On simultaneous rational approximations to a real number, its square and its cube, Acta Artih. 133 (2008), 185-197.
    MathSciNet     CrossRef

  16. D. Roy, On Schmidt and Summerer parametric geometry of numbers, arXiv: 1406.3669.

  17. D. Roy, Construction of points realizing the regular systems of Wolfgang Schmidt and Leonhard Summerer, arXiv:1406.3669

  18. D. Roy, Spectrum of the exponents of best rational approximation, arXiv:1410.1007.

  19. J. Schleischitz, Diophantine approximation and special Liouville numbers, Commun. Math. 21 (2013), 39-76.
    MathSciNet    

  20. J Schleischitz, Two estimates concerning classical diophantine approximation constants, Publ. Math. Debrecen, 84 (2014), 415-437.
    MathSciNet     CrossRef

  21. J. Schleischitz, On the spectrum of Diophantine approximation constants, Mathematika, to appear.

  22. J. Schleischitz, Generalizations of a result of Jarnik on simultaneous approximation, arXiv: 1410.6697.

  23. W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526-551.
    MathSciNet     CrossRef

  24. W. M. Schmidt and L. Summerer, Parametric geometry of numbers and applications, Acta Arith. 140 (2009), 67-91.
    MathSciNet     CrossRef

  25. W. M. Schmidt and L. Summerer, Diophantine approximation and parametric geometry of numbers, Monatsh. Math. 169 (2013), 51-104.
    MathSciNet     CrossRef

  26. W. M. Schmidt and L. Summerer, Simultaneous approximation to three numbers, Mosc. J. Comb. Number Theory 3 (2013), 84-107.
    MathSciNet    

  27. V. G. Sprindzuk, A proof of Mahler's conjecture on the measure of the set of S-numbers, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 379-436. English translation in: Amer. Math. Soc. Transl. 51 (1966), 215-272.
    MathSciNet    

  28. R. C. Vaughan and S. Velani, Diophantine approximation on planar curves: the convergence theory, Invent. Math. 166 (2006), 103-124.
    MathSciNet     CrossRef

  29. M. Waldschmidt, Recent advances in Diophantine approximation, Number theory, analysis and geometry, Springer, New York, 2012, p. 659-704
    MathSciNet    

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