The fractional dimensional theory of continued fractions

Abstract

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.

Hausdorff dimension of the set of singular pairs

“…Implicit in this terminology is the expectation that the set Sing * (d) of all nondegenerate singular vectors is somehow larger than the union of all rational hyperplanes in R d , which is a set of Hausdorff dimension d − 1. The papers [1], [24], [19] and [2] give lower bounds on certain subsets of Sing Divergent trajectories. There is a well-known dynamical interpretation of singular vectors.…”

“…Good [7] that includes many of the basic features of the main calculation except for the use of accelerations. .…”

Hausdorff dimension of real numbers with bounded digit averages

“…Finally, Lemma 6 provides a characterization of the Hausdorff dimension which involves covers formed by fundamental intervals of variable depth. Then Lemma 7 (which extends Lemma 7 of [17]) shows that the Hausdorff dimension is completely characterized by covers of fixed depth.…”

“…When the set A is "open", there exists a (unique) real s = τ A for which λ A (s) = 1 and the Hausdorff dimension of E A equals τ A . The particular case of "constrained" continued fractions was extensively studied; the beginners were Jarník [28,29], Besicovitch [5] and Good [17]. Then Cusick [11], Hirst [24] and Bumby [9] brought important contributions, and finally Hensley [19][20][21][22] completely solved the problem.…”

Chaotic and topological properties of continued fractions

“…In 1845, Gauss observed that T preserves the probability measure given by Continued fractions is a kind of representation of real numbers and an important tool to study the Diophantine approximation in number theory. Many metric and dimensional results on Diophantine approximation have been obtained with the help of continued fractions such as Good [5], Jarnik [7], Khintchine [9] etc, and also the dimensional properties of continued fractions were considered, for example Wang and Wu [21], Xu [26] etc. The behaviors of the continued fraction dynamical systems have been widely investigated, for example, the shrinking target problems [13], mixing property [18], thermodynamic formalism [15], limit theorems [3] etc.…”