Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems
Mariusz Urbański1 ; Anna Zdunik2
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
Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 261-286.

We prove that if by Jn(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 limn→∞Hhn(Jn(𝒢))=1=H1(J(𝒢)), where hn is the Hausdorff dimension of Jn(𝒢), Hhn 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 lim̲F→EHhF(JF)<Hh𝒮(J𝒮); the lower limit is taken over finite subsets of the countable infinite alphabet E.

Nous montrons que, si Jn(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 limn→∞Hhn(Jn(𝒢))=1=H1(J(𝒢)), où hn est la dimension de Hausdorff de Jn(𝒢), Hhn 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 lim̲F→EHhF(JF)<Hh𝒮(J𝒮) ; cette limite inférieure s’étend sur les sous-ensembles finis de l’alphabet infini E.

Reçu le :
Accepté le :
Publié le :
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, Tome 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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