The average length of finite continued fractions with fixed denominator

Bykovskii, V. A.; Frolenkov, Dmitry

doi:10.1070/sm8718

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Asymptotics For the Number Of Steps Of Euclidean Algorithms1

Introduction1

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2022

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Cited by 1 publication

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“…here A 1 is as before and A 2 is also an explicitly given non-zero constant. Bykovskiȋ and Frolenkov [6] have recently obtained a generalisation of this and obtained an improved error term.…”

Section: Asymptotics For the Number Of Steps Of Euclidean Algorithmsmentioning

confidence: 91%

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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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Section: Asymptotics For the Number Of Steps Of Euclidean Algorithmsmentioning

confidence: 91%

“…2 For the same average, an asymptotic formula with two significant terms was obtained later by Porter [ 21 ]: here is as before and is also an explicitly given non-zero constant. Bykovskiĭ and Frolenkov [ 6 ] have recently obtained a generalisation of this and obtained an improved error term.…”

Section: Introductionmentioning

confidence: 99%