KD
K. Dajani
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We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques—induced transformations and contraction—in the setting of Shunji Ito's ([19]) natural extension of the Farey tent map, which generates ‘slow’ continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's S-expansions, and Nakada's parameterised family of α-continued fractions for all 0<α≤1 as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the α-continued fraction transformations as an explicit induced transformation of Ito's natural extension.
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We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques—induced transformations and contraction—in the setting of Shunji Ito's ([19]) natural extension of the Farey tent map, which generates ‘slow’ continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's S-expansions, and Nakada's parameterised family of α-continued fractions for all 0<α≤1 as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the α-continued fraction transformations as an explicit induced transformation of Ito's natural extension.
We revisit Ito’s [Osaka J. Math. 26 (1989), pp. 557–578] natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results—including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of L´evy’s Theorem to subsequences of convergents and mediants—are presented as corollaries within this unifying theory.
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We revisit Ito’s [Osaka J. Math. 26 (1989), pp. 557–578] natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results—including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of L´evy’s Theorem to subsequences of convergents and mediants—are presented as corollaries within this unifying theory.
We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).
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We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).