## Euclidean algorithms are gaussian (2006)

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Venue: | Journal of Number Theory, Volume 110, Issue |

Citations: | 23 - 10 self |

### BibTeX

@ARTICLE{Baladi06euclideanalgorithms,

author = {Viviane Baladi and Brigitte Vallée},

title = {Euclidean algorithms are gaussian},

journal = {Journal of Number Theory, Volume 110, Issue},

year = {2006},

volume = {110},

pages = {331--386}

}

### OpenURL

### Abstract

Abstract. We obtain a Central Limit Theorem for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence. We also provide very precise asymptotic estimates and error terms for the mean and variance of such parameters. For costs that are lattice (including the number of steps), we go further and establish a Local Limit Theorem, with optimal speed of convergence. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasi-powers theorems, and the saddle-point method. Such dynamical analyses had previously been used to perform the average-case analysis of algorithms. For the present (dynamical) analysis in distribution, we require estimates on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics [16]. 1.

### Citations

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(Show Context)
Citation Context ...eigenvalue λ = 1, and a spectral gap: the rest of the spectrum lies in a disk of radius < 1. For a non constant digit-cost satisfying a moderate growth condition (2.5), elementary perturbation theory =-=[28]-=- implies that H1,w inherits the spectral gap when w is near 0. This gives the above-mentioned quasi-powers expansion and, togethersEUCLIDEAN ALGORITHMS ARE GAUSSIAN 3 with convexity of the logarithm o... |

401 | Analytic Combinatorics
- Flajolet, Sedgewick
- 2008
(Show Context)
Citation Context ...(2s,w) and the quasi-inverse of Hs,w.6 VIVIANE BALADI AND BRIGITTE VALLÉE In a sense, distributional analyses are obtained by (uniform) perturbation of average-case analyses, as Flajolet explains in =-=[24]-=-: “Parameters of combinatorial structures, provided they remain ‘simple’ enough, lead to local deformations (via an auxiliary variable w considered near 0) of the functional relations defining univari... |

351 |
Universal codeword sets and representations of the integers
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- 1975
(Show Context)
Citation Context ...and will be encoded as 1111 0 0101. The process is quite easy to decode. For instance, the word 00011111100001000101 is decoded as (1,1,1,68,1,3). We encounter there a code that was proposed by Elias =-=[21]-=- for encoding integers. We can also adopt the Fano-Shannon principle to the case of the other two Algorithms, even if the existence of the sign ǫ makes the process more involved.EUCLIDEAN ALGORITHMS ... |

280 | An Introduction to the Analysis of Algorithms
- Sedgewick, Flajolet
- 1995
(Show Context)
Citation Context ...al cost C of the rational trajectories. The expectation EN[C] is described by the partial sums of the coefficients of a generating function S(s) (a common tool in the average-case study of algorithms =-=[19, 20]-=-) where the parameter s “marks” the size N of inputs. As it is usual in number theory, the generating functions S(s) are Dirichlet series. Recently, Vallée [49] has related S(2s) to the quasi-inverse ... |

227 |
Thermodynamic Formalism
- Ruelle
- 1978
(Show Context)
Citation Context ... and Mayer [34]). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, in connection with his thermodynamic formalism (see e.g. =-=[41]-=-). We shall see that “weighted” transfer operators appear naturally in the probabilistic analysis of dynamics. Fix a reference probability measure on [0, 1], absolutely continuous with a smooth densit... |

222 |
Introduction to analytic and probabilistic number theory
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- 1995
(Show Context)
Citation Context ...half-plane {ℜs > 1}, and analytic on ℜs = 1 except for a simple pole at s = 1. Under these conditions, one can extract asymptotically the coefficients of S(s) by means of Delange’s Tauberian theorems =-=[14, 44]-=-. (Just like in the usual prime number theorem, or in weighted dynamical prime number theorems, see e.g. Sections 6–7 of [37].) For costs of moderate growth and Euclidean algorithms in the Fast Class,... |

190 |
Harmonic Analysis. Real variable methods, orthogonality and oscillatory integrals
- Stein
- 1993
(Show Context)
Citation Context ...rh,k(x)dx, I exp[itΨh,k(x)] Rh,k(x)dx involves Ψh,k, Rh,k defined I with rh,k defined in (3.9). We shall apply the following lemma to each oscillatory integral I(h, k): Van der Corput Lemma (See e.g. =-=[43]-=-). For each interval I and every Q > 0, there is C(Q), so that for all t ∈ R, Ψ ∈ C2 (I) with |Ψ ′′ (x)| ≤ Q , |Ψ ′ (x)| ≥ ∆ with |t| −1 ≤ ∆ ≤ 1, and r ∈ C1 (I) with ||r||0 ≤ R , ||r||1,1 ≤ RD, the in... |

179 |
Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérisque
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- 1990
(Show Context)
Citation Context ...ally the coefficients of S(s) by means of Delange’s Tauberian theorems [14, 44]. (Just like in the usual prime number theorem, or in weighted dynamical prime number theorems, see e.g. Sections 6–7 of =-=[37]-=-.) For costs of moderate growth and Euclidean algorithms in the Fast Class, this dynamical approach gives [49] that the mean value EN[C] of the total cost of the rational trajectory satisfies EN[C] ∼ ... |

130 |
Continued Fractions
- Khinchin
- 1997
(Show Context)
Citation Context ...he density transformer, also known as the Perron-Frobenius operator, (1.4) H1[f](x) = � |h ′ (x)| · f ◦ h(x) h∈H was introduced early in the study of continued fractions (see e.g. Lévy [33], Khinchin =-=[29]-=-, Kuzmin [31], Wirsing [51], Babenko [3], and Mayer [34]). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, in connection wi... |

101 |
Perturbation Theory of Linear Operators, Springer:New York-BerlinHeidelberg
- Kato
- 1976
(Show Context)
Citation Context ...). As soon as a supplementary property is fulfilled, namely Property Anw(1,0) expressing that the transfer operator H1,w is analytic with respect to w when w is near 0, elementary perturbation theory =-=[27]-=- entails that the two properties UDE and SG extend to the operator H1,w for w near 0. In the case when parameters are af moderate growth, this leads to a proof that the asymptotic distribution of tota... |

85 |
Positive transfer operators and decay of correlations
- Baladi
- 2000
(Show Context)
Citation Context ...es of the transfer operator Hs,w. Endow the Banach space C1 (I) with the norm � · �1,1. It is known that for (ℜs, ℜw) ∈ Σ0 × W0 the operator Hs,w is bounded but not compact acting on C1 (I) [see e.g. =-=[4, 9]-=-]; however, it is quasi-compact. We recall the definition of quasi-compactness for a bounded operator L on a Banach space: Denote by SpL the spectrum of L, by R(L) its spectral radius, and by Re(L) it... |

73 |
Exact Real Computer Arithmetic with Continued Fractions
- Vuillemin
- 1990
(Show Context)
Citation Context ...oduced by the Euclidean Algorithm. Many applications use CF expansions in an extensive way: for instance, it is possible to make computations on rational numbers by using directly their CF expansions =-=[6, 55]-=-. It is thus important to encode efficiently the sequence of CF-digits so that this encoding can be directly used in further computations, and in order to to compare the average length of these two co... |

67 | On convergence rates in the central limit theorems for combinatorial structures
- Hwang
- 1998
(Show Context)
Citation Context ... [1] for a more abstract framework and references to the pioneering paper of Nagaev. It is convenient to base the proof below on a compact and versatile statement, the “Quasi-Powers Theorem” of Hwang =-=[25, 26, 27]-=- [Theorem 0 below], which encapsulates the consequences of the Lévy continuity theorem and the Berry-Esseen inequality. Continued fractions of rational numbers: Average-case analysis of Euclidean algo... |

62 |
On decay of correlations in Anosov flows
- Dolgopyat
- 1998
(Show Context)
Citation Context ...on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics =-=[16]-=-. 1. Introduction According to Knuth [30, p. 335], “we might call Euclid’s method the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.”... |

55 |
Ergodic Theory of Fibred Systems and Metric Number Theory
- Schweiger
- 1995
(Show Context)
Citation Context ...nitely many branches, and is just Renyi’s condition otherwise. To check that maps associated to our algorithms are in the good class, for σ0 = 1/2, use |h ′ [m,ǫ] | = O(m−2 ). (See also Figure 1, and =-=[42]-=- for proofs.) If I is endowed with an initial probability density g0 with respect to Lebesgue measure, T acts on it and transforms it into a new density g1. The operator H such that g1 = H[g0] is call... |

49 |
Dynamical Zeta Functions for Piecewise Monotone
- Ruelle
- 1994
(Show Context)
Citation Context ..., it is defined as H[f](x) := ∑ |h ′ (x)| f ◦ h(x) h∈H and its iterate of order n satisfies an analogous relation, but now with respect to the set Hn , H n [f](x) := ∑ |h ′ (x)| f ◦ h(x) h∈H n Ruelle =-=[40, 41]-=- shows that it proves useful to deal with a more general operator, the transfer operator, that depends on some complex parameter s and is denoted by Hs. It satisfies Hs[f](x) := ∑ |h ′ (x)| s f ◦ h(x)... |

47 |
Théoremes limites pour les structures combinatoires et les fonctions arithmetiques
- Hwang
- 1994
(Show Context)
Citation Context ...imit Theorem, see for instance [13] or [10] for interval maps, and [1] for a more abstract framework. The proof we give in Section 2 is based on a compact and versatile QuasiPowers statement of Hwang =-=[28, 29, 30]-=- (Theorem 0) which encapsulates the consequences of the two main tools usually applied to get the CLT with speed of convergence: the Lévy continuity theorem and the BerryEessen inequality. Theorem 0 i... |

39 |
An Introduction to Chaotic Dynamical Systems Second Edition
- Devaney
- 1989
(Show Context)
Citation Context ...) := P(u,v) ∑ i=1 c(hi). We next see how to associate to each algorithm a dynamical system of the interval, i.e., a transformation T : I → I. For nice and elementary surveys on interval dynamics, see =-=[8, 35, 14, 18]-=-. The interval I has already been defined (see Figure 3) and T extends the map defined on rationals by the equality T(u/v) = r/u where r is the remainder of the Euclidean division on (u,v). It is easy... |

35 | Some limit theorems for stationary Markov chains - Nagaev - 1957 |

34 |
Limit theorems for partially hyperbolic systems
- Dolgopyat
(Show Context)
Citation Context ... Figure 2. Properties of the transfer operator that are needed for analysing trajectories. [35, 36], Pollicott and Sharp [37] used them to find error terms in some asymptotic estimates, and Dolgopyat =-=[16]-=- deduced a class of limit theorems for dynamical systems. To best of our knowledge, this paper is the first instance where these powerful tools are applied for distributional analyses in discrete comb... |

32 |
Continued fractions and related transformations
- Mayer
- 1991
(Show Context)
Citation Context ...us operator, (1.4) H1[f](x) = � |h ′ (x)| · f ◦ h(x) h∈H was introduced early in the study of continued fractions (see e.g. Lévy [33], Khinchin [29], Kuzmin [31], Wirsing [51], Babenko [3], and Mayer =-=[34]-=-). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, in connection with his thermodynamic formalism (see e.g. [41]). We shall... |

31 | Large Deviations for Combinatorial Distributions I
- Hwang
- 1996
(Show Context)
Citation Context ... [1] for a more abstract framework and references to the pioneering paper of Nagaev. It is convenient to base the proof below on a compact and versatile statement, the “Quasi-Powers Theorem” of Hwang =-=[25, 26, 27]-=- [Theorem 0 below], which encapsulates the consequences of the Lévy continuity theorem and the Berry-Esseen inequality. Continued fractions of rational numbers: Average-case analysis of Euclidean algo... |

30 |
Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps
- Aaronson, Denker
(Show Context)
Citation Context ...tion expansion of the form 1 (1.1) x = . 1 m1 + 1 m2 + . .. 1 + mn + . . . Ordinary continued fraction expansions can be viewed as trajectories of a onedimensional dynamical system, the Gauss map T : =-=[0, 1]-=- → [0, 1], T(x) := 1 − ⌊1 ⌋, for x �= 0, T(0) = 0 . x x (Here, ⌊x⌋ is the integer part of x.) For an irrational x, the trajectory T (x) = (x, T(x), T 2 (x), . . . , T n (x), . . .) never meets 0 and i... |

30 |
Transformations dilatantes de l’intervalle et théorèmes limites. d’opérateurs de transfert et applications. Astérisque 238
- Broise
- 1996
(Show Context)
Citation Context ...ution of the total cost is Gaussian, with (optimal) speed of convergence O(1/ √ n). This Central Limit Theorem [stated below more precisely as Theorem 1] is quite well-known. See for instance [13] or =-=[9]-=- for interval maps, and [1] for a more abstract framework and references to the pioneering paper of Nagaev. It is convenient to base the proof below on a compact and versatile statement, the “Quasi-Po... |

30 | Dynamical Sources in Information Theory : Fundamental intervals and Word Prefixes
- Vallée
(Show Context)
Citation Context ...that H ∗ 1 preserves �µ1, gives the expressions as integrals. To finish, apply Rohlin’s formula. (6.b) Convexity of the pressure is an old theme, see, e.g., [41] and also [37], Chapter 4, Prop. 10 in =-=[45]-=-, Prop. 3.8 in [11], Proposition 6.1 in [9]. We adapt here the last work. It is stated in the context of functions with bounded variation. Due to our strong Markov assumption, we may work in C 1 (I). ... |

29 | Analysis of the binary Euclidean algorithm
- Brent
- 1976
(Show Context)
Citation Context ...I − Hs,w). Open problems. We also ask various questions about Euclidean algorithms: for instance, what happens for other Euclidean algorithms of the Fast Class (in particular for the Binary algorithm =-=[8, 46]-=-)? The extension of our results to cost functions that are still “small” but take into account the boolean cost (also known as the bit complexity) of each arithmetic operation is on our agenda. Note t... |

29 |
The number of steps in the Euclidean algorithm
- Hensley
- 1994
(Show Context)
Citation Context ...r to all these questions for three different algorithms that all belong to the Fast Class. Concerning the standard Euclidean algorithm and the number of steps (i.e., the constant cost c ≡ 1), Hensley =-=[24]-=- has obtained a Central Limit Theorem, and a Local Limit Theorem with speed of convergence O((log N) −1/24 ). In the present work, we apply dynamical methods for the first time to the distributional a... |

27 | Central limit theorem and stable laws for intermittent maps, preprint
- Gouëzel
- 2003
(Show Context)
Citation Context ... known to be possible via operator techniques [2, 49]. On another register, the extension to “large” costs is likely to lead us to the realm of stable laws: see for instance Gouezel and Vardi’s works =-=[21, 50]-=- for occurrences of these laws in continued fraction related matters. Acknowledgements. During his thesis, Hervé Daudé made experiments providing evidence for the Gaussian limit property of the number... |

24 | Exponential error terms for growth functions on negatively curved manifolds
- Pollicott, Sharp
- 1998
(Show Context)
Citation Context ...ater on, Pollicott and Sharp used Dolgopyat’s bounds together with Perron’s formula to find error terms in asymptotic estimates for geodesic flows on surfaces of variable negative curvature; see e.g. =-=[38]-=-, where only univariate Dirichlet series with positive cofficients appear. To the best of our knowledge, the present paper is the first instance where these powerful tools are extended to dynamical sy... |

24 | A note on the theory of moment generating functions - Curtiss - 1942 |

24 | The thermodynamic formalism approach to Selberg’s zeta function for PSL(2
- Mayer
- 1991
(Show Context)
Citation Context ...ty of Perron’s formula is the “uniform” Property UEVL on the norm of iterates of the transfer operator Hs,w when (ℜs, ℜw) is near the reference point (1,0) (see Figure 1). Note that the work of Mayer =-=[39, 40]-=- relative to the standard Euclidean algorithm shows that there is α > 0 for which the quasi inverse (I − Hs,0) −1 is analytic for ℜs ≥ 1 − α and s = 1. However, this property does not suffice and we ... |

23 |
Continued fractions. The University of Chicago
- Khinchin
- 1964
(Show Context)
Citation Context ...as Perron–Frobenius operator), encapsulates all the important informations relative to the ”dynamics” of the iterative process. Within the realm of continued fractions, studies of Lévy [33], Khinchin =-=[28]-=-, Kuzmin [30], Wirsing [51], Babenko [2], and Mayer [34] describe the main properties of the density transformer. Two properties of the density transformer, namely, the existence of a unique dominant ... |

22 |
The Number of Steps in the Euclidean Algorithm
- Dixon
- 1970
(Show Context)
Citation Context ...ber of steps P(u, v), which corresponds to the trivial digit-cost c ≡ 1, the standard Euclidean algorithm was first analyzed in the average-case around 1969, independently by Heilbronn [22] and Dixon =-=[15]-=-. The centered algorithm was studied by Rieger [39]. Consider now a general digit-cost c of moderate growth and the associated total cost C of the rational trajectories. The expectation EN[C] is descr... |

22 |
On the thermodynamic formalism for the Gauss map
- Mayer
- 1990
(Show Context)
Citation Context ...ty of Perron’s formula is the “uniform” Property UEVL on the norm of iterates of the transfer operator Hs,w when (ℜs, ℜw) is near the reference point (1,0) (see Figure 1). Note that the work of Mayer =-=[39, 40]-=- relative to the standard Euclidean algorithm shows that there is α > 0 for which the quasi inverse (I − Hs,0) −1 is analytic for ℜs ≥ 1 − α and s = 1. However, this property does not suffice and we ... |

21 |
Markov approximations and decay of correlations for Anosov
- Chernov
- 1998
(Show Context)
Citation Context ... By the above discussion, such bounds would follow from similar estimates on the quasi-inverse of the operator, which are closely related to those obtained by Dolgopyat [16]. In the spirit of Chernov =-=[12]-=-, Dolgopyat [16] introduced several “uniform nonintegrability” (UNI) conditions. They allowed him to control oscillatory integrals associated to iterates of transfer operators Hs for s = σ + it, with ... |

21 |
Dynamics of the Binary Euclidean Algorithm: Functional Analysis
- Vallée
- 1998
(Show Context)
Citation Context ...ion: (2.13) EN[exp(wC)] := Φw(N) Φ0(N) , where Φw(N) = Φc,w(N) is the cumulative value of exp[wC] on ΩN, (2.14) Φw(N) := � exp[wC(u, v)] , Φ0(N) = |ΩN | . (u,v)∈ΩN Extending the principles defined in =-=[46, 48, 49]-=-, we replace the sequence of moment generating functions by a single bivariate Dirichlet series, henceforth called the Dirichlet-moment generating function: (2.15) S(s, w) := � 1 � cn(w) exp[wC(u, v)]... |

17 | Dynamical analysis of a class of Euclidean algorithms, Theoret
- Vallée
(Show Context)
Citation Context ...rictly positive. (7) [Function w ↦→ σ(w).] There is a complex neighborhood W of 0 and a unique function σ : W → C such that λ(σ(w), w) = 1, this function is analytic, and σ(0) = 1. Proof. We refer to =-=[4, 9, 48]-=- except for claim (7) (which follows from λ ′ (1) �= 0 and the implicit function theorem) and for claim (6): (6.a) Taking the derivatives at (1, 0) of Hs,w[fs,w] = λ(s, w)fs,w (with respect to s or w)... |

16 | Average bit–complexity of Euclidean Algorithms
- Akhavi, Vallée
(Show Context)
Citation Context ...to account the boolean cost (also known as the bit complexity) of each arithmetic operation is on our agenda. Note that an averagecase analysis is already known to be possible via operator techniques =-=[2, 49]-=-. On another register, the extension to “large” costs is likely to lead us to the realm of stable laws: see for instance Gouezel and Vardi’s works [21, 50] for occurrences of these laws in continued f... |

16 |
les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue
- Lévy, Sur
- 1929
(Show Context)
Citation Context ...of this fact. The density transformer, also known as the Perron-Frobenius operator, (1.4) H1[f](x) = � |h ′ (x)| · f ◦ h(x) h∈H was introduced early in the study of continued fractions (see e.g. Lévy =-=[33]-=-, Khinchin [29], Kuzmin [31], Wirsing [51], Babenko [3], and Mayer [34]). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, i... |

16 |
Laws of chaos: invariant measures and dynamical systems in one dimension
- Boyarsky, Gora
- 1997
(Show Context)
Citation Context ...us extensions. It proves rewarding to study continued fractions algorithms in the dynamical systems framework. We consider here dynamical systems of the interval. For nice and elementary surveys, see =-=[7, 4, 31]-=-. A dynamical system (of the interval) (I,T) consists of an interval I together with a continuous map T : I → I. which is called the shift mapping. Given an initial condition x in I, the sequence (x,T... |

15 |
On the average length of a class of continued fractions, Number Theory and Analysis
- Heilbronn
- 1969
(Show Context)
Citation Context ...xt. For the number of steps P(u, v), which corresponds to the trivial digit-cost c ≡ 1, the standard Euclidean algorithm was first analyzed in the average-case around 1969, independently by Heilbronn =-=[22]-=- and Dixon [15]. The centered algorithm was studied by Rieger [39]. Consider now a general digit-cost c of moderate growth and the associated total cost C of the rational trajectories. The expectation... |

15 |
On the Theorem of Gauss-Kusmin-Lévy and a Frobenius-type Theorem for Function Space.” Acta Arithmetica 24
- Wirsing
- 1974
(Show Context)
Citation Context ...o known as the Perron-Frobenius operator, (1.4) H1[f](x) = � |h ′ (x)| · f ◦ h(x) h∈H was introduced early in the study of continued fractions (see e.g. Lévy [33], Khinchin [29], Kuzmin [31], Wirsing =-=[51]-=-, Babenko [3], and Mayer [34]). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, in connection with his thermodynamic formal... |

14 |
The art of Computer programming, Volume 2, 3rd edition
- Knuth
- 1998
(Show Context)
Citation Context ...IAN 37 the right shape of the error term (O(N −γ )), for which Porter further showed that one could take γ = 1 6 −ǫ in the case of the standard algorithm. We refer the reader to the accounts by Knuth =-=[30]-=- and Finch [18] for more material on this classical topic. Our formula (6.1) also extends Rieger’s analyses (first published around 1980, see [39, 40]) of the centered algorithm and the odd algorithm.... |

14 | Digits and continuants in Euclidean algorithms. Ergodic versus Tauberian theorems
- Vallée
(Show Context)
Citation Context ...tions of rational numbers: Average-case analysis of Euclidean algorithms. There are variants of the standard continued fraction algorithm induced by variations of the standard division procedure. See =-=[49]-=- for many examples and a classification into the “Fast” and the “Slow” Class. In this paper, we study three algorithms of the Fast Class, the standard, centered, and odd algorithms, specified at the b... |

12 |
Some ergodic properties of maps of the interval, in: Dynamical systems
- Collet
- 1996
(Show Context)
Citation Context ... distribution of the total cost is Gaussian, with (optimal) speed of convergence O(1/ √ n). This Central Limit Theorem [stated below more precisely as Theorem 1] is quite well-known. See for instance =-=[13]-=- or [9] for interval maps, and [1] for a more abstract framework and references to the pioneering paper of Nagaev. It is convenient to base the proof below on a compact and versatile statement, the “Q... |

10 |
Généralisation du théorème d’Ikehara
- Delange
- 1954
(Show Context)
Citation Context ...half-plane {ℜs > 1}, and analytic on ℜs = 1 except for a simple pole at s = 1. Under these conditions, one can extract asymptotically the coefficients of S(s) by means of Delange’s Tauberian theorems =-=[14, 44]-=-. (Just like in the usual prime number theorem, or in weighted dynamical prime number theorems, see e.g. Sections 6–7 of [37].) For costs of moderate growth and Euclidean algorithms in the Fast Class,... |

9 |
On a problem of
- Babenko
- 1978
(Show Context)
Citation Context ... Perron-Frobenius operator, (1.4) H1[f](x) = � |h ′ (x)| · f ◦ h(x) h∈H was introduced early in the study of continued fractions (see e.g. Lévy [33], Khinchin [29], Kuzmin [31], Wirsing [51], Babenko =-=[3]-=-, and Mayer [34]). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, in connection with his thermodynamic formalism (see e.g.... |

8 |
O.: Sur un problème de Gauss. Atti del Congresso Internazionale dei Matematici, Bologna 6
- Kuzmin
- 1928
(Show Context)
Citation Context ...ansformer, also known as the Perron-Frobenius operator, (1.4) H1[f](x) = � |h ′ (x)| · f ◦ h(x) h∈H was introduced early in the study of continued fractions (see e.g. Lévy [33], Khinchin [29], Kuzmin =-=[31]-=-, Wirsing [51], Babenko [3], and Mayer [34]). The density transformer is a special case of a transfer operator. Very general transfer operators were introduced by Ruelle, in connection with his thermo... |

7 | Dynamical analysis of α-Euclidean algorithms
- Bourdon, Daireaux, et al.
(Show Context)
Citation Context ...2 (0), where h∗ is the mirror LFT of h, defined by: h ∗ (x) = ax + c bx + d if h(x) = ax + b cx + d . This mirror operation appears in [42] where Schweiger relates it to the natural extension, and in =-=[7]-=-, where the authors use the geometric notion of “folded” and “unfolded.” Clearly, the mirror map is an involution satisfying the morphism property (h ◦ k) ∗ = k ∗ ◦h ∗ . It is not difficult to see tha... |

7 |
Über die mittlere Schrittanzahl bei Divisionsalgorithmen
- Rieger
- 1978
(Show Context)
Citation Context ...ial digit-cost c ≡ 1, the standard Euclidean algorithm was first analyzed in the average-case around 1969, independently by Heilbronn [22] and Dixon [15]. The centered algorithm was studied by Rieger =-=[39]-=-. Consider now a general digit-cost c of moderate growth and the associated total cost C of the rational trajectories. The expectation EN[C] is described by the partial sums of the coefficients of a g... |