# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.