On evil Kronecker sequences and lacunary trigonometric products
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 637-687

An important result of Weyl states that for every sequence ${\left({n}_{k}\right)}_{k\ge 1}$ of distinct positive integers the sequence of fractional parts of ${\left({n}_{k}\alpha \right)}_{k\ge 1}$ is u.d. mod 1 for almost all $\alpha$. However, in this general case it is usually extremely difficult to measure the speed of convergence of the empirical distribution of $\left(\left\{{n}_{1}\alpha \right\},\cdots ,\left\{{n}_{N}\alpha \right\}\right)$ towards the uniform distribution. In this paper we investigate the case when ${\left({n}_{k}\right)}_{k\ge 1}$ is the sequence of evil numbers, that is the sequence of non-negative integers having an even sum of digits in base 2. We utilize a connection with lacunary trigonometric products ${\prod }_{\ell =0}^{L}\left|sin\pi {2}^{\ell }\alpha \right|$, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of ${\left({n}_{k}\alpha \right)}_{k\ge 1}$ and for the discrepancy of ${\left(\left\{{n}_{k}\alpha \right\}\right)}_{k\ge 1}.$ Furthermore, we provide some explicit examples of numbers $\alpha$ for which we can give estimates for the discrepancy of ${\left(\left\{{n}_{k}\alpha \right\}\right)}_{k\ge 1}$.

Un résultat important de Weyl nous dit que pour chaque suite ${\left({n}_{k}\right)}_{k\ge 1}$ de nombres entiers positifs différents la suite ${\left\{{n}_{k}\alpha \right\}}_{k\ge 1}$ est équidistribuée modulo $1$ pour presque tous les réels $\alpha$. Dans ce cas, il est d’habitude extrêmement difficile de mesurer la vitesse de convergence de la distribution empirique vers l’équidistribution.

Dans cet article, nous étudions le cas ou ${\left({n}_{k}\right)}_{k\ge 1}$ est la suite des nombres entiers « méchants », donc la suite des nombres positifs la une somme de chiffres paire dans la base 2. Nous relions ce probléme aux produits trigonométriques ${\prod }_{l=0}^{L}\parallel sin\pi {2}^{l}\alpha \parallel$ en donnant des estimations exactes pour de tels produits et nous obtenons des estimations exactes pour la discrépance de la suite ${\left\{{n}_{k}\alpha \right\}}_{k\ge 1}$.

En plus, nous donnons des exemples concrets de réels $\alpha$ pour lesquels nous pouvons obtenir des estimations pour la discrépance de la suite ${\left\{{n}_{k}\alpha \right\}}_{k\ge 1}$.

Received : 2016-01-11
Revised : 2016-04-07
Accepted : 2016-05-12
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3094
Classification:  11B85,  11K38,  11B83,  11A63,  68R15
Keywords: evil numbers, Thue–Morse sequence, $\left(n\alpha \right)$-sequence, discrepancy, lacunary trigonometric products
@article{AIF_2017__67_2_637_0,
author = {Aistleitner, Christoph and Hofer, Roswitha and Larcher, Gerhard},
title = {On evil Kronecker sequences and lacunary trigonometric products},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {2},
year = {2017},
pages = {637-687},
doi = {10.5802/aif.3094},
language = {en},
url = {http://www.numdam.org/item/AIF_2017__67_2_637_0}
}

Aistleitner, Christoph; Hofer, Roswitha; Larcher, Gerhard. On evil Kronecker sequences and lacunary trigonometric products. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 637-687. doi : 10.5802/aif.3094. http://www.numdam.org/item/AIF_2017__67_2_637_0/

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