On the discrepancy of sequences associated with the sum-of-digits function
Annales de l'Institut Fourier, Volume 37 (1987) no. 3, p. 1-17

If $w=\left({q}_{k}{\right)}_{k\in \mathbf{N}}$ denotes the sequence of best approximation denominators to a real $\alpha$, and ${s}_{\alpha }\left(n\right)$ denotes the sum of digits of $n$ in the digit representation of $n$ to base $w$, then for all $x$ irrational, the sequence $\left({s}_{\alpha }\left(n\right)·x{\right)}_{n\in \mathbf{N}}$ is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if $\alpha$ has bounded continued fraction coefficients.

Soit $\left[{a}_{0};{a}_{1}...\right]$ le développement en fraction continue du nombre irrationnel $\alpha$ ; soit $w=\left({q}_{k}\right)$ la suite de dénominateur des réduites successives de $\alpha$. Tout entier naturel $n$ se développe de manière unique sous la forme $n=\Sigma {\epsilon }_{k}\left(n\right){q}_{k}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{3.33333pt}{0ex}}{s}_{\alpha }\left(n\right)=\Sigma {\epsilon }_{k}\left(n\right)$ est la somme de chiffres de $n$. La suite $\left(x{s}_{\alpha }\left(n\right){\right)}_{n\in \mathbf{N}}$ est équirépartie modulo 1 si $x$ est irrationnel. Nous prouvons quelques estimations de la discrépance de la suite $\left(x{s}_{\alpha }\left(n\right){\right)}_{n\in \mathbf{N}}$.

@article{AIF_1987__37_3_1_0,
author = {Larcher, Gerhard and Kopecek, N. and Tichy, R. F. and Turnwald, G.},
title = {On the discrepancy of sequences associated with the sum-of-digits function},
journal = {Annales de l'Institut Fourier},
publisher = {Institut Fourier},
volume = {37},
number = {3},
year = {1987},
pages = {1-17},
doi = {10.5802/aif.1095},
zbl = {0601.10038},
mrnumber = {89c:11119},
language = {en},
url = {http://www.numdam.org/item/AIF_1987__37_3_1_0}
}

Larcher, Gerhard; Kopecek, N.; Tichy, R. F.; Turnwald, G. On the discrepancy of sequences associated with the sum-of-digits function. Annales de l'Institut Fourier, Volume 37 (1987) no. 3, pp. 1-17. doi : 10.5802/aif.1095. http://www.numdam.org/item/AIF_1987__37_3_1_0/

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