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.

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 $\left(x{s}_{\alpha }\left(n\right){\right)}_{n\in \mathbf{N}}$.

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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://archive.numdam.org/articles/10.5802/aif.1095/

[1] J. Coquet, Représentation des entiers naturels et suites uniformément équiréparties, Ann. Inst. Fourier, 32-1 (1982), 1-5. | Numdam | MR | Zbl

[2] J. Coquet, Répartition de la somme des chiffres associée à une fraction continue, Bull. Soc. Roy. Liège, 52 (1982), 161-165. | MR | Zbl

[3] J. Coquet, G. Rhin, Ph. Toffin, Représentations des entiers naturels et indépendance statistique 2, Ann. Inst. Fourier, 31-1 (1981), 1-15. | Numdam | MR | Zbl

[4] E. Hlawka, Theorie der Gleichverteilung, Bibl. Inst. Mannheim-Wien-Zürich, 1979. | MR | Zbl

[5] H. Kawai, α-additive Functions and Uniform Distribution modulo one, Proc. Japan. Acad. Ser. A., 60 (1984), 299-301. | MR | Zbl

[6] J.F. Koksma, Some theorems on diophantine inequalities, Math. Centrum Amsterdam, Scriptum no. 5, 1950. | MR | Zbl

[7] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley and Sons, New York, 1974. | MR | Zbl

[8] W.M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201. | MR | Zbl

[9] R.F. Tichy and G. Turnwald, on the discrepancy of some special sequences, J. Number Th., 26 (1987), 68-78. | MR | Zbl

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