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 (n k ) k≥1 of distinct positive integers the sequence of fractional parts of (n k α) k≥1 is u.d. mod 1 for almost all α. However, in this general case it is usually extremely difficult to measure the speed of convergence of the empirical distribution of ({n 1 α},⋯,{n N α}) towards the uniform distribution. In this paper we investigate the case when (n k ) k≥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 ∏ ℓ=0 L sinπ2 ℓ α, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of n k α k≥1 and for the discrepancy of n k α k≥1 . Furthermore, we provide some explicit examples of numbers α for which we can give estimates for the discrepancy of n k α k≥1 .
Un résultat important de Weyl nous dit que pour chaque suite (n k ) k≥1 de nombres entiers positifs différents la suite {n k α} k≥1 est équidistribuée modulo 1 pour presque tous les réels α. 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 (n k ) k≥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 ∏ l=0 L ∥sinπ2 l α∥ en donnant des estimations exactes pour de tels produits et nous obtenons des estimations exactes pour la discrépance de la suite {n k α} k≥1 .En plus, nous donnons des exemples concrets de réels α pour lesquels nous pouvons obtenir des estimations pour la discrépance de la suite {n k α} k≥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, (nα)-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 = {https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/item/AIF_2017__67_2_637_0/

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