On the frequency of partial quotients of regular continued fractions

Ai-Hua Fan (1, 2) , Lingmin Liao (1, 2) , Ji-Hua Ma (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 Department of Mathematics [Wuhan]

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

Ai-Hua Fan, Lingmin Liao, Ji-Hua Ma. On the frequency of partial quotients of regular continued fractions. Mathematical Proceedings of the Cambridge Philosophical Society, 2010, 148, pp.179--192. ⟨hal-00396379⟩