A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets

Hensley, Doug

doi:10.1006/jnth.1996.0058

Cited by 75 publications

(68 citation statements)

References 14 publications

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“…Nous les avons adopte s aux probleÁ mes contraints. Notons que Hensley a, de manieÁ re inde pendante, effectue des calculs similaires dans le cas particulier d'une contrainte e le mentaire finie [He4]. L'ide e du travail pre ce dent est la suivante: on``approche'' l'action de l'ope rateur G s sur des fonctions analytiques par son action sur les polyno^mes de Taylor de ces fonctions en un point a de l'intervalle [0,1].…”

Section: Des Motivations Et Des Exemples Nume Riquesunclassified

Dynamique des fractions continues à contraintes périodiques

Vallée

1998

Journal of Number Theory

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On e tudie ici, dans un cadre ge ne ral, les nombres re els, rationnels, ou irrationnels quadratiques dont le de veloppement en fraction continue obe it aÁ un ensemble de contraintes non ne cessairement fini mais pe riodique. Plus pre cise ment, on de termine la dimension de Hausdorff de l'ensemble des re els contraints, la densite des rationnels et des irrationnels quadratiques contraints. Puis, on analyse le comportement moyen de l'algorithme d'Euclide sur ces rationnels contraints ainsi que la longueur moyenne de la pe riode des irrationnels quadratiques contraints et la valeur moyenne de la constante de Le vy des irrationnels quadratiques contraints. En associant un ope rateur qui engendre cet ensemble de contraintes, on relie tous les objets e tudie s aux proprie te s spectrales dominantes de cet ope rateur. En ce qui concerne la dimension de Hausdorff et la densite des rationnels contraints, les re sultats pre sente s e tendent aÁ des contraintes quelconques des re sultats pre ce dents obtenus uniquement dans le cas de contraintes e le mentaires. Les autres e tudes densite des irrationnels quadratiques contraints; comportement moyen de la hauteur et de la pe riode d'un de veloppement en fraction continue contraint; valeur moyenne de la constante de Le vy des irrationnels quadratiques contraints semblent nouvelles me^me dans le cas d'une contrainte e le mentaire. Academic Press

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Section: Des Motivations Et Des Exemples Nume Riquesunclassified

Dynamique des fractions continues à contraintes périodiques

Vallée

1998

Journal of Number Theory

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“…Bourgain-Kontorovich give A 0 = 50 as an allowable value for A, and this has since been reduced to A 0 = 5 [FK14,Hua15]; furthermore, Hensley [Hen96] has conjectured that A 0 = 2 is allowable, and that the error rate O(e −c √ log N ) can 9 For the application to be immediate, Γ would need to be a Zariski-dense group and not just a semi-group; minor modifications are needed to handle this case.…”

Section: James Conjecturementioning

confidence: 99%

Schubert calculus and torsion explosion

Williamson

2016

J. Amer. Math. Soc.

101

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Abstract. We observe that certain numbers occurring in Schubert calculus for SLn also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. These numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig's conjecture on the characters of simple rational modules for SLn over fields of positive characteristic. Our examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.

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“…These "dynamical" methods have already been used in other domains of algorithmic number theory, mainly in the study of the classical Euclidean algorithm, or related algorithms [Ma2], [Hen1]- [Hen4], [Va1]- [Va3], [DFV]. In particular, the author [FV] uses them to obtain an alternative proof of the results of Heilbronn and Dixon about the logarithmic behavior of the classical Euclidean algorithm.…”

Section: Binary Euclidean Algorithm (U V)mentioning

confidence: 99%

Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators

Vallée

1998

Algorithmica

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Abstract. We provide here a complete average-case analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N . We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of the binary Euclidean algorithm, and finally the sum of the exponents of powers of 2 contained in the numerators of the binary continued fraction. The average values of these parameters are shown to be asymptotic to A i log N , and the three constants A i are related to the invariant measure of the Perron-Frobenius operator linked to this dynamical system. The binary Euclidean algorithm was previously studied in 1976 by Brent who provided a partial analysis of the number of steps, based on a heuristic model and some unproven conjecture. Our methods are quite different, not relying on unproven assumptions, and more general, since they allow us to study all the parameters of the binary continued fraction expansion.

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A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets

Cited by 75 publications

References 14 publications

Dynamique des fractions continues à contraintes périodiques

Dynamique des fractions continues à contraintes périodiques

Schubert calculus and torsion explosion

Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators

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