### A generalization of a theorem of Lekkerkerker to Ostrowski's decomposition of natural numbers

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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.

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...

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...

We prove that the sums ${S}_{k}$ of independent random vectors satisfy $P(ma{x}_{1\le k\le n}\parallel {S}_{k}\parallel >3t)\le 2ma{x}_{1\le k\le n}P(\parallel {S}_{k}\parallel >t)$, t ≥ 0.

For any positive integer $m$ let $F\left(m\right)$ 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 $F\left(m\right)$.