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Numbers with Almost all Convergents in a Cantor Set

Part of: Elementary number theory Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  03 December 2018

Damien Roy  and
Johannes Schleischitz
Damien Roy
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur, Ottawa, ON, K1N 6N5 Email: droy@uottawa.cajohannes.schleischitz@univie.ac.at
Johannes Schleischitz
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur, Ottawa, ON, K1N 6N5 Email: droy@uottawa.cajohannes.schleischitz@univie.ac.at
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Abstract

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In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.

Keywords

Cantor setcontinued fractionDiophantine approximationparametric geometry of numbers

MSC classification

Primary: 11A55: Continued fractions
Secondary: 11J25: Diophantine inequalities 11J82: Measures of irrationality and of transcendence
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 869 - 875
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The research of J. Schleischitz was supported by the Schrödinger Scholarship J 3824 of the Austrian Science Fund (FWF), while that of D. Roy was partly supported by an NSERC discovery grant.

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