On certain statistical properties of continued fractions with even and with odd partial quotients

Boca, Florin P.; Vandehey, Joseph

doi:10.4064/aa156-3-1

Cited by 9 publications

(9 citation statements)

References 14 publications

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“…The following statement ([6, Proposition 3.2], see also [20]) will be useful. Note that the formulation in [6] on the interval [0, 1] is different than the one given here on [1, ∞).…”

Section: Even Continued Fractions and The Pell Equationmentioning

confidence: 99%

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

Boca,

Siskaki

2020

Preprint

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This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (Rényi). We show that they coincide exactly with two sets of quadratic irrationals that we call E-reduced, and respectively B-reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.

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“…The following statement ([6, Proposition 3.2], see also [20]) will be useful. Note that the formulation in [6] on the interval [0, 1] is different than the one given here on [1, ∞).…”

Section: Even Continued Fractions and The Pell Equationmentioning

confidence: 99%

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

Boca,

Siskaki

2020

Preprint

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“…Thus, h 1 is a rational point with planar Siegel coordinates h 1 = r (1) q (1) , p (1) q (1) . Furthermore, we have q (1) = −p, so that…”

Section: Rational Pointsmentioning

confidence: 99%

Continued fractions on the Heisenberg group

Lukyanenko

Vandehey

2015

Acta Arith.

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Abstract. We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions. We then discuss dynamical properties of the associated Gauss map, comparing them with base-b expansions on the Heisenberg group and continued fractions on the complex plane.

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“…В частности, это позволяет -исследовать типичное поведение алгоритмов Евклида с округлением до ближайшего целого [48], [49], с четными и нечетными неполными частными [50]- [52], при делении с избытком [53], [54], при делении вычитанием [55], [56] и типичное поведение диагональных дробей Минковского [57] и цепных дробей более общего вида [58], а также изучать разложение в цепные дроби квадратичных рациональностей [45];…”

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Three-dimensional continued fractions and Kloosterman sums

Устинов¹,

Ustinov²

2015

Uspekhi Mat. Nauk

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Обзор посвящен результатам, связанным с метрическими свойствами классических цепных дробей и трехмерных цепных дробей Вороного-Минковского. Основное внимание уделяется применению аналитических методов, основанных на оценках сумм Клостермана. В статье развивается аппарат, предназначенный для решения задач на трехмерных решетках. В основе подхода лежит идея редукции к предыдущей размерности, применявшаяся ранее Линником и Скубенко при исследовании целочисленных решений детерминантного уравнения det X = P , где X-матрица размера 3 × 3 с независимыми коэффициентами и P-растущий параметр. Предлагаемый метод применяется для изучения статистических свойств трехмерных цепных дробей Вороного-Минковского в решетках с фиксированным определителем. В частности, для среднего числа базисов Минковского доказывается асимптотическая формула со степенным понижением в остаточном члене. Этот результат можно считать трехмерным аналогом теоремы Портера о средней длине конечных цепных дробей. Библиография: 127 названий.

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On certain statistical properties of continued fractions with even and with odd partial quotients

Abstract: We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for Nakada's α-expansions.

Cited by 9 publications

References 14 publications

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

Continued fractions on the Heisenberg group

Three-dimensional continued fractions and Kloosterman sums

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