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Hausdorff dimensions of bounded-type continued fraction sets of Laurent series. (English) Zbl 1112.11037
Continued fraction expansion and Diophantine approximation in fields of formal Laurent series have been studied by many researchers. S. Kristensen [Math. Proc. Camb. Philos. Soc. 135, No. 2, 255–268 (2003; Zbl 1088.11056)] has studied Hausdorff dimension and Khinchin-type theorems in the field of Laurent series.
In this paper the author studies the Hausdorff dimensions of bounded-type continued fraction sets of Laurent series and show that the Texan conjecture is valid in the case of Laurent series.

MSC:
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
11J61 Approximation in non-Archimedean valuations
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[1] Artin, E., Quadratische Körper im gebiete der höheren kongruenzen, I-II, math. Z., 19, 153-246, (1924) · JFM 51.0144.05
[2] Berthé, V.; Nakada, H., On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Exposition math., 18, 4, 257-284, (2000) · Zbl 1024.11050
[3] Bumby, R.T., Hausdorff dimensions of Cantor sets, J. reine angew. math., 331, 192-206, (1982) · Zbl 0468.10032
[4] R.T. Bumby, Hausdorff Dimension of Sets Arising in Number Theory, Number Theory, New York, 1983-84, Lecture Notes in Mathematics, vol. 1135, Springer, Berlin, 1985, pp. 1-8. · Zbl 0575.28004
[5] T.W. Cusick, M.E. Flahive, The Markoff and Lagrange Spectra, vol. 30, AMS Mathematical Surveys and Monographs, Providence, RI, 1989. · Zbl 0685.10023
[6] Falconer, K.J., Fractal geometry, mathematical foundations and application, (1990), Wiley New York
[7] Good, I.J., The fractional dimension theory of continued fractions, Proc. Cambridge philos. soc., 37, 199-228, (1941) · JFM 67.0988.03
[8] Hensley, D., The Hausdorff dimensions of some continued fraction Cantor sets, J. number theory, 33, 2, 182-198, (1989) · Zbl 0689.10060
[9] Hensley, D., A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. number theory, 58, 1, 9-45, (1996) · Zbl 0858.11039
[10] Jenkinson, O., On the density of Hausdorff dimensions of bounded type continued fraction sets: the texan conjecture, Stochastic dynamics, 4, 1, 63-76, (2004) · Zbl 1089.28006
[11] Jenkinson, O.; Pollicott, M., Computing the dimension of dynamically defined sets: \(E_2\) and bounded continued fractions, Ergodic theory dynamical systems, 21, 5, 1429-1445, (2001) · Zbl 0991.28009
[12] Kristensen, S., On well-approximable matrices over a field of formal series, Math. proc. Cambridge philos. soc., 135, 2, 255-268, (2003) · Zbl 1088.11056
[13] Lasjaunias, A., A survey of Diophantine approximation in fields of power series, Monatsh. math., 130, 3, 211-229, (2000) · Zbl 0990.11043
[14] Mauldin, R.D.; Urbański, M., Dimensions and measures in infinite iterated function systems, Proc. London math. soc., 73, 3, 105-154, (1996) · Zbl 0852.28005
[15] Mauldin, R.D.; Urbański, M., Conformal iterated function systems with applications to the geometry of continued fractions, Trans. amer. math. soc., 351, 12, 4995-5025, (1999) · Zbl 0940.28009
[16] H. Niederreiter, The probabilistic theory of linear complexity, in: C.G. Günther (Ed.), Advances in Cryptology—EUROCRYPT’88, Lecture Notes in Computer Science, vol. 330, 1988, pp. 191-209.
[17] Niederreiter, H.; Vielhaber, M., Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles, J. complexity, 13, 3, 353-383, (1997) · Zbl 0934.94013
[18] Schmidt, W.M., On continued fractions and Diophantine approximation in power series fields, Acta arith., 95, 139-166, (2000) · Zbl 0987.11041
[19] Shallit, J., Real numbers with bounded partial quotients: a survey, Enseign. math., 38, 2, 151-187, (1992) · Zbl 0753.11006
[20] Wu, J., On the sum of degrees of digits occurring in continued fraction expansions of Laurent series, Math. proc. Cambridge philos. soc., 138, 1, 9-20, (2005) · Zbl 1062.11054
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