ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

doi:10.1016/S0252-9602(13)60053-3

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (4): 943-949.doi: 10.1016/S0252-9602(13)60053-3

ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

胡动刚|胡学海*

Huazhong Agricultural University, Wuhan 430070, China

收稿日期:2012-07-17 出版日期:2013-07-20 发布日期:2013-07-20
通讯作者: Xuehai HU, College of Science, Huazhong Agricultural University. E-mail:hudg@mail.hzau.edu.cn; huxuehai@mail.hzau.edu.cn
基金资助:
Xuehai HU was supported by NSFC (11001093, 10901066).

ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

HU Dong-Gang, HU Xue-Hai*

Huazhong Agricultural University, Wuhan 430070, China

Received:2012-07-17 Online:2013-07-20 Published:2013-07-20
Contact: Xuehai HU, College of Science, Huazhong Agricultural University. E-mail:hudg@mail.hzau.edu.cn; huxuehai@mail.hzau.edu.cn
Supported by:
Xuehai HU was supported by NSFC (11001093, 10901066).

摘要/Abstract

摘要：

A famous theorem of Szemer’edi asserts that any subset of integers with posi-tive upper density contains arbitrarily arithmetic progressions. Let F_q be a finite field with q elements and F_q((X⁻¹)) be the power field of formal series with coefficients lying in F_q. In this paper, we concern with the analogous Szemer´edi problem for continued fractions of Laurent series: we will show that the set of points x ∈ F_q((X−1)) of whose sequence
of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.

关键词: Szemer´edi theorem, continued fractions, Laurent series, Hausdorff dimension

Abstract:

Key words: Szemer´edi theorem, continued fractions, Laurent series, Hausdorff dimension

中图分类号:

11K55

胡动刚, 胡学海. ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES[J]. 数学物理学报(英文版), 2013, 33(4): 943-949.

HU Dong-Gang, HU Xue-Hai. ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES[J]. Acta mathematica scientia,Series B, 2013, 33(4): 943-949.

参考文献

[1] Artin E. Quadratische K¨orper im Gebiete der H¨oheren Kongruenzen I-II. Math Z, 1924, 19: 153–246

[2] Lasjaunias A. A survey of Diophantine approximation in fields of power series. Monatsh Math, 2000, 130: 211–229

[3] Schmidt W M. On continued fractions and Diophantine approximation in power series fields. Acta Arith, 2000, 95: 139–166

[4] Berth´e V, Nakada H. On continued fraction expansions in positive characteristic: equivalence relations and some metric properties. Expo Math, 2000, 18: 257–284

[5] Kristensen S. On well-approximable matrices over a field of formal series. Math Proc Cambridge Philos Soc, 2003, 135: 255–268

[6] Hu X H, Wang B W, Wu J, Yu Y L. Cantor sets determined by partial quotients of continued fractions of Laurent series. Finite Fields Appl, 2008, 14: 417–437

[7] Wu J. On the sum of degrees of digits occurring in continued fraction expansions of Laurent series. Math Proc Camb Philo Soc, 2005, 138: 9–20

[8] Wu J. Hausdorff dimensions of bounded type continued fraction sets of Laurent series. Finite Fields Appl, 2007, 13: 20–30

[9] Niederreiter H. The probabilistic theory of linear complexity//G¨unther C G. Advances in Cryptology-EUROCRYPT’88. Lecture Note in Computer Science 330. New York: Springer-Verlag, 1988: 191–209

[10] Niederreiter H, Vielhaber M. Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles. J Complexity, 1997, 13: 353–383

[11] Van der Waerden B L. Beweis einer Baudetschen Vermutung. Nieuw Arch Wisk, 1927, 15: 212–216

[12] Szemer´edi E. On sets of integers containing no k elements in arithmetic progression. Acta Arith, 1975, 27: 299–345

[13] Furstenberg H. Ergodic behavior of diagonal measures and a theorem of Szemer’edi on arithmetic progressions. J Anal Math, 1977, 31: 204–256

[14] Green B, Tao T. The primes contain arbitrarily long arithmetic progressions. Ann Math, 2008, 167: 481–547

[15] Feng D J, Wu J. The Hausdorff dimension of recurrent sets in symbolic spaces. Nonlinearity, 2001, 14: 81–85

[16] Tong X, Wang B W. How many points contain arithmetic progressions in their continued fraction expansion? Acta Arith, 2009, 139: 369–376

[17] Falconer K J. Techniques in Fractal Geometry. Wiley, 1997.

ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

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