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Arithmetic diophantine approximation for continued fractions-like maps on the interval

Avraham Bourla (2014)

Acta Arithmetica

We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of Möbius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fraction expansions.

Digits and continuants in euclidean algorithms. Ergodic versus tauberian theorems

Brigitte Vallée (2000)

Journal de théorie des nombres de Bordeaux

We obtain new results regarding the precise average-case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the average-case complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide...

Efficient computation of addition chains

F. Bergeron, J. Berstel, S. Brlek (1994)

Journal de théorie des nombres de Bordeaux

The aim of this paper is to present a unifying approach to the computation of short addition chains. Our method is based upon continued fraction expansions. Most of the popular methods for the generation of addition chains, such as the binary method, the factor method, etc..., fit in our framework. However, we present new and better algorithms. We give a general upper bound for the complexity of continued fraction methods, as a function of a chosen strategy, thus the total number of operations required...

Products and quotients of numbers with small partial quotients

Stephen Astels (2002)

Journal de théorie des nombres de Bordeaux

For any positive integer m let F ( m ) denote the set of numbers with all partial quotients (except possibly the first) not exceeding m . In this paper we characterize most products and quotients of sets of the form F ( m ) .

Řetězové zlomky s předepsanou periodou

Martin Kuděj (2021)

Pokroky matematiky, fyziky a astronomie

Článek se zabývá řetězovými zlomky algebraických čísel stupně 2. Jsou ukázány jejich základní vlastnosti, včetně potřebné teorie. Ta je poté použita k nalezení tvaru řetězového zlomku druhých odmocnin z přirozených čísel, které nejsou čtverce, a jejich symetrické posloupnosti ( a 1 , ... , a k ) . Dále, pro danou symetrickou posloupnost přirozených čísel ( a 1 , ... , a k ) jsou charakterizována všechna přirozená čísla, jejichž druhé odmocniny mají řetězový zlomek právě s touto symetrickou posloupností. Tato přirozená čísla jsou popsána...

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