A generalization of the Jarnik-Besicovitch theorem by continued fractions

作者：Wang, Bao-Wei; Wu, Jun; Xu, Jian^*

来源：Ergodic Theory and Dynamical Systems, 2016, 36(04): 1278-1306.

DOI：10.1017/etds.2014.98

• ITERATED FUNCTION SYSTEMS
• SIMULTANEOUS DIOPHANTINE APPROXIMATION
• HAUSDORFF DIMENSION
• RATIONAL NUMBERS
• CANTOR SETS
• CONJECTURE
• EXPANSIONS
• DIGIT
• ORDER

We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarnik-Besicovitch theorem, localized Jarnik-Besicovitch theorem and its several generalizations. As is well known, the classic Jarnik-Besicovitch sets, expressed in terms of continued fractions, can be written as
{x is an element of[0,1) : a(n+1)(x) >= e(tau(log vertical bar T'x vertical bar+...+logT'(Tn-1x)vertical bar)) for for infinitely many n is an element of N}.
where T is the Gauss map and a(n)(x) is the nth partial quotient of x. In this paper, we consider the size of the generalized Jarnik-Besicovitch set
{x is an element of[0,1) : a(n+1)(x) >= e(tau(x)(f(x)+...+f(Tn-1x)vertical bar)) for for infinitely many n is an element of N}.
where tau (x) and f(x) are positive functions defined on T [0, 1].

• 出版日期2016-6
• 单位华中科技大学