Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems
Journal de Théorie des Nombres de Bordeaux, Volume 28 (2016) no. 1, pp. 261-286.

We prove that if by J n (𝒢) we denote the set of all numbers in [0,1] whose infinite continued fraction expansions have all entries in the finite set {1,2,...,n}, then lim n H h n (J n (𝒢))=1=H 1 (J(𝒢)), where h n is the Hausdorff dimension of J n (𝒢), H h n is the corresponding Hausdorff measure, and J(𝒢) denotes the set of all irrational numbers in [0,1], i .e. those whose continued fraction expansion is infinite. We also show that this property is not too common by constructing a class of infinite iterated function systems 𝒮 on [0,1], consisting of similarities, for which lim ̲ FE H h F (J F )<H h 𝒮 (J 𝒮 ); the lower limit is taken over finite subsets of the countable infinite alphabet E.

Nous montrons que, si J n (𝒢) est l’ensemble des réels dans [0,1] dont la fraction continue infinie est constituée de nombres entiers compris entre 1 et n, alors lim n H h n (J n (𝒢))=1=H 1 (J(𝒢)), où h n est la dimension de Hausdorff de J n (𝒢), H h n est la mesure de Hausdorff correspondant et où J(𝒢) est l’ensemble de tous les nombres irrationnels de [0,1], i.e. ceux dont la fraction continue est infinie. Nous montrons aussi que cette propriété n’est pas générale en construisant une classe de systèmes de fonctions itérées 𝒮 sur [0,1], formés de similarités, pour lesquels lim ̲ FE H h F (J F )<H h 𝒮 (J 𝒮 ) ; cette limite inférieure s’étend sur les sous-ensembles finis de l’alphabet infini E.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.938
Classification: 11A55,  28A78,  28A80,  37D35
Keywords: Continued fractions, Hausdorff measure, Gauss map, bounded distortion, iterated function systems
Mariusz Urbański 1; Anna Zdunik 2

1 Department of Mathematics University of North Texas Denton, TX 76203-1430 USA
2 Institute of Mathematics University of Warsaw ul. Banacha 2, 02-097 Warszawa POLAND
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Mariusz Urbański; Anna Zdunik. Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems. Journal de Théorie des Nombres de Bordeaux, Volume 28 (2016) no. 1, pp. 261-286. doi : 10.5802/jtnb.938. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.938/

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