On evil Kronecker sequences and lacunary trigonometric products

Aistleitner, Christoph; Hofer, Roswitha; Larcher, Gerhard

Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 637-687.

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 ${\left\{n_k\alpha \right\}}_{k\ge 1}$.

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 $(\left\{n_1\alpha \right\},\cdots ,\left\{n_N\alpha \right\})$ 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 ${\left(\left\{n_k\alpha \right\}\right)}_{k\ge 1}$.

Classification : 11B85, 11K38, 11B83, 11A63, 68R15
Keywords: evil numbers, Thue–Morse sequence, $(n\alpha )$-sequence, discrepancy, lacunary trigonometric products
Mot clés : nombres entiers méchants, suites de Thue–Morse, suites $(n\alpha )$, discrépancie, produits trigonométriques

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