Acta mathematica scientia,Series B ›› 2013, Vol. 33 ›› Issue (4): 943-949.doi: 10.1016/S0252-9602(13)60053-3

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ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

 HU Dong-Gang, HU Xue-Hai*   

  1. 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).

Abstract:

A famous theorem of Szemer’edi asserts that any subset of integers with posi-tive upper density contains arbitrarily arithmetic progressions. Let Fq be a finite field with q elements and Fq((X−1)) be the power field of formal series with coefficients lying in Fq. 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 ∈ Fq((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.

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

CLC Number: 

  • 11K55
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