A remark on continued fractions with sequences of partial quotients

作者：Wu Jun^*

来源：Journal of Number Theory, 2008, 128(8): 2394-2397.

DOI：10.1016/j.jnt.2007.12.004

• continued fraction
• partial quotient
• Hausdorff dimension

Given any infinite set B of positive integers b(1) < b(2) < ... , let tau(B) denote the exponent of convergence of the series Sigma(infinity)(n=1) b(n)(-s). Let E(B) be the set {x epsilon [0, 1]: a(n)(x) epsilon B (n >= 1) and a(n)(x) -> infinity as n -> infinity). Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973) 221-227] proved the inequality dim(H) E(B) <= tau(B)/2 and conjectured (see Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973), p. 225] and Cusick [T.W. Cusick, Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford Ser. (2) 41 (1990), p. 278]) that equality holds in general. In [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press], we gave a positive answer to this conjecture. In this note, we further show that the result in [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press] is sharp.

• 出版日期2008-8
• 单位华中科技大学