Results 1  10
of
24
Continued Fraction Algorithms, Functional Operators, and Structure Constants
, 1996
"... Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer o ..."
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Cited by 28 (4 self)
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Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer operators  due to Ruelle and Mayer (also following Lévy, Kuzmin, Wirsing, Hensley, and others) that describe precisely the dynamics of the continued fraction transformation. Spectral characteristics of transfer operators are shown to have many consequences, like the normal law for logarithms of continuants associated to the basic continued fraction algorithm and a purely analytic estimation of the average number of steps of the Euclidean algorithm. Transfer operators also lead to a complete analysis of the "Hakmem" algorithm for comparing two rational numbers via partial continued fraction expansions and of the "digital tree" algorithm for completely sorting n real numbers by means of ...
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 22 (10 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 17 (4 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... 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 averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 14 (5 self)
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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 averagecase 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 a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
A distributional limit law for continued fraction digit sums
, 2006
"... We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this proce ..."
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Cited by 5 (4 self)
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We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalized linearly we determine a large deviation asymptotic.
The complexity of generating functions for integer points in polyhedra and beyond
 In Proceedings of the International Congress of Mathematicians
, 2006
"... Abstract. Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ Z d of integer points for which an a priori “long ” Laurent series or polynomial � m∈S xm can be written as a “short ” rational function f(S; x). Examples include the sets of integer point ..."
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Cited by 4 (0 self)
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Abstract. Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ Z d of integer points for which an a priori “long ” Laurent series or polynomial � m∈S xm can be written as a “short ” rational function f(S; x). Examples include the sets of integer points in rational polyhedra, integer semigroups, and Hilbert bases of rational cones, among others. We discuss applications to efficient counting and optimization and open questions.
LARGE DEVIATION ASYMPTOTICS FOR CONTINUED FRACTION EXPANSIONS
, 2007
"... ABSTRACT. We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation process. Also a large deviation asym ..."
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Cited by 2 (1 self)
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ABSTRACT. We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation process. Also a large deviation asymptotic for single digits is given. 1.
Recent results on the periodic Lorentz gas
 arXiv:0906.0191. LINEAR FLOW IN A HONEYCOMB 735
"... Abstract. The DrudeLorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption — known as the Boltzm ..."
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Abstract. The DrudeLorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption — known as the BoltzmannGrad scaling by analogy with the kinetic theory of rarefied gases — this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J.
Euclidean dynamics
 Discrete and Continuous Dynamical Systems
"... Abstract. We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as tran ..."
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Cited by 2 (1 self)
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Abstract. We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various tools of analytic combinatorics: generating functions, Dirichlet series, Tauberian theorems, Perron’s formula and quasipowers theorems. Such dynamical analyses can be used to perform the averagecase analysis of algorithms, but also (dynamical) analysis in distribution. 1. Introduction. Computing the Greatest Common Divisor [Gcd
Limiting Curlicue Measures for Theta Sums
, 2009
"... We consider the ensemble of curves {γα,N: α ∈ (0, 1], N ∈ N} obtained by linearly interpo ..."
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Cited by 1 (0 self)
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We consider the ensemble of curves {γα,N: α ∈ (0, 1], N ∈ N} obtained by linearly interpo