On the law of the iterated logarithm for trigonometric series with bounded gaps II

Christoph Aistleitner; Katusi Fukuyama

doi:10.5802/jtnb.945

Christoph Aistleitner ¹ ; Katusi Fukuyama ¹

¹ Department of Mathematics Graduate School of Science Kobe University, Kobe 657-8501, Japan

Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 2, pp. 391-416.

Résumé
Abstract

Dans un article précédent, nous avons montré l’existence d’une suite ${(n_{k})}_{k \geq 1}$ de nombres entiers positifs avec sauts bornés telle que nous ayons une loi logarithm itérée (LIL) de la forme

\underset{N \to \infty}{lim sup} \frac{|\sum_{k = 1}^{N} cos 2 π n_{k} x|}{\sqrt{N log log N}} = \infty pour presque tous x .

Dans le présent travail, nous prouvons un résultat complémentaire montrant que tout comportement limsup prescrit dans la LIL est possible pour des suites avec sauts bornés. Plus précisément, nous montrons que pour tout nombre réel $Λ \geq 0$ , il existe une suite d’entiers ${(n_{k})}_{k \geq 1}$ satisfaisant $n_{k + 1} - n_{k} \in {1, 2}$ telle que la limsup dans la LIL soit égale à $Λ$ pour presque tout $x$ . Des résultats similaires sont montrés pour des sommes $\sum f (n_{k}, x)$ et pour la discrépance ${(〈 n_{k} x 〉)}_{k \geq 1}$ .

In an earlier paper we proved that there exists a sequence ${(n_{k})}_{k \geq 1}$ of positive integers with bounded gaps such that we have a law of the iterated logarithm (LIL) in the form

\underset{N \to \infty}{lim sup} \frac{|\sum_{k = 1}^{N} cos 2 π n_{k} x|}{\sqrt{N log log N}} = \infty for almost all x .

In the present paper we prove a complementary results showing that any prescribed limsup-behavior in the LIL is possible for sequences with bounded gaps. More precisely, we show that for any real number $Λ \geq 0$ there exists a sequence of integers ${(n_{k})}_{k \geq 1}$ satisfying $n_{k + 1} - n_{k} \in {1, 2}$ such that the limsup in the LIL equals $Λ$ for almost all $x$ . Similar results are proved for sums $\sum f (n_{k} x)$ and for the discrepancy of ${(〈 n_{k} x 〉)}_{k \geq 1}$ .

Reçu le : 2014-03-25
Révisé le : 2014-09-11
Accepté le : 2014-10-16
Publié le : 2017-01-02

MR Zbl

DOI : 10.5802/jtnb.945

Classification : 60F15, 11K38, 42A32
Mots-clés : Law of the iterated logarithm, lacunary trigonometric series, metric discrepancy theory, probabilistic methods

Affiliations des auteurs :

Christoph Aistleitner ¹ ; Katusi Fukuyama ¹

¹ Department of Mathematics Graduate School of Science Kobe University, Kobe 657-8501, Japan

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Christoph Aistleitner; Katusi Fukuyama. On the law of the iterated logarithm for trigonometric series with bounded gaps II. Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 2, pp. 391-416. doi : 10.5802/jtnb.945. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.945/

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