Weak multipliers for generalized van der Corput sequences

Pausinger, Florian

doi:10.5802/jtnb.819

Pausinger, Florian ¹

¹ IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400-Klosterneuburg, Austria

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 729-749.

Résumé
Abstract

Les suites de Van der Corput généralisées sont des suites unidimensionnelles et infinies dans l’intervalle de l’unité. Elles sont générées par permutations des entiers de la base $b$ et sont les éléments constitutifs des suites multi-dimensionnelles de Halton. Suites aux progrès récents d’Atanassov concernant le comportement de distribution uniforme des suites de Halton nous nous intéressons aux permutations de la formule $P (i) = a i (mod b)$ pour les entiers premiers entre eux $a$ et $b$ . Dans cet article nous identifions des multiplicateurs $a$ générant des suites de Van der Corput ayant une mauvaise distribution. Nous donnons les bornes inférieures explicites pour cette distribution asymptotique associée à ces suites et relions ces dernières aux suites générées par permutation d’identité, qui sont, selon Faure, les moins bien distribuées des suites généralisées de Van der Corput dans une base donnée.

Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base $b$ and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form $P (i) = a i (mod b)$ for coprime integers $a$ and $b$ . We show that multipliers $a$ that either divide $b - 1$ or $b + 1$ generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences.

MR Zbl

DOI : 10.5802/jtnb.819

Classification : 11K06, 11K38
Mots-clés : Uniform distribution, diaphony, generalized van der Corput sequence

Affiliations des auteurs :

Pausinger, Florian ¹

¹ IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400-Klosterneuburg, Austria

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     title = {Weak multipliers for generalized van der {Corput} sequences},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {729--749},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     doi = {10.5802/jtnb.819},
     zbl = {1270.11075},
     mrnumber = {3010637},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.819/}
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Pausinger, Florian. Weak multipliers for generalized van der Corput sequences. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 729-749. doi : 10.5802/jtnb.819. https://www.numdam.org/articles/10.5802/jtnb.819/

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