Partial quotients and equidistribution

Chang, Mei-Chu

doi:10.1016/j.crma.2011.06.007

Number Theory

Partial quotients and equidistribution
[Fractions continues et equidistribution]

Présenté par : Jean Bourgain

Chang, Mei-Chu ¹

¹ Department of Mathematics, University of California, 900 University Avenue, Riverside, CA 92521, USA

Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 713-718.

Résumé
Abstract

Nous obtenons des bornes en moyenne pour les quotients partiels de certaines fractions $b / p$ , p un nombre premier, b dans un sous-groupe de ${(Z / p Z)}^{⁎}$ ainsi que pour b un élément primitif « typique » $(\mod p)$ . Ceci donne en particulier une amélioration de résultats de G. Larcher. Il est bien connu que le comportement des quotients partiels de $\frac{b}{p}$ détermine les propriétés statistiques de la distribution $b^{j} (\mod p)$ . On en déduit, comme corollaire, de meilleures estimations sur les corrélations partielles pour ces suites.

We establish average bounds on the partial quotients of fractions $b / p$ , with p prime, b taken in a multiplicative subgroup of ${(Z / p Z)}^{⁎}$ and for “most” primitive elements b. Our result improves upon earlier work due to G. Larcher. The behavior of the partial quotients of $b / p$ is well known to be crucial to the statistical properties of the pseudo-congruential number generator $(\mod p)$ . As a corollary, estimates on their pair correlation are refined.

Reçu le : 2011-04-15
Accepté le : 2011-06-07
Publié le : 2011-06-24

DOI : 10.1016/j.crma.2011.06.007

Affiliations des auteurs :

Chang, Mei-Chu ¹

¹ Department of Mathematics, University of California, 900 University Avenue, Riverside, CA 92521, USA

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Chang, Mei-Chu. Partial quotients and equidistribution. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 713-718. doi : 10.1016/j.crma.2011.06.007. http://www.numdam.org/articles/10.1016/j.crma.2011.06.007/

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