Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms

Frolenkov, Dmitry

doi:10.1070/sm2012v203n02abeh004223

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…and the value of C 3 being given by a somewhat longer, yet similar expression which we omit here. Both error terms in (1.5) and (1.6) have been improved to O((log Q) 3 /Q) by Frolenkov [8] who incorporated ideas of Selberg from the elementary proof of the prime number theorem.…”

Section: Figurementioning

confidence: 99%

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Minelli¹,

Sourmelidis²,

Technau³

2022

Preprint

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We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval (0, 1/2), establishing that they behave differently on (0, 1/2) than they do on (1/2, 1). These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients.As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums.The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovskiȋ and others, ultimately dating back to Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.

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Section: Figurementioning

confidence: 99%

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Minelli¹,

Sourmelidis²,

Technau³

2022

Preprint

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“…and the value of C 3 being given by a somewhat longer, yet similar expression which we omit here. Both error terms in (1.5) and (1.6) have been improved to O((log Q) 3 /Q) by Frolenkov [10] who incorporated ideas of Selberg from the elementary proof of the prime number theorem.…”

Section: Asymptotics For the Number Of Steps Of Euclidean Algorithmsmentioning

confidence: 99%

“…This was improved by Zhabitskaya [ 30 ] (following the approach of Ustinov [ 24 ]), a few years later, who showed that where are explicitly given non-zero constants, the first two being given by and the value of being given by a somewhat longer, yet similar expression which we omit here. Both error terms in ( 1.5 ) and ( 1.6 ) have been improved to by Frolenkov [ 10 ] who incorporated ideas of Selberg from the elementary proof of the prime number theorem.…”

Section: Introductionmentioning

confidence: 99%

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

2022

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We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval (0, 1/2), establishing that they behave differently on (0, 1/2) than they do on (1/2, 1). These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovskiĭ and others, ultimately dating back to Lochs and Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.

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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.

show abstract

Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms

Cited by 4 publications

References 15 publications

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Three-dimensional continued fractions and Kloosterman sums

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