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=(q k ) kN denotes the sequence of best approximation denominators to a real α, and s α (n) denotes the sum of digits of n in the digit representation of n to base w, then for all x irrational, the sequence (s α (n)·x) nN is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if α has bounded continued fraction coefficients.

Soit [a 0 ;a 1 ...] le développement en fraction continue du nombre irrationnel α ; soit w=(q k ) la suite de dénominateur des réduites successives de α. Tout entier naturel n se développe de manière unique sous la forme n=Σε k (n)q k ;s α (n)=Σε k (n) est la somme de chiffres de n. La suite (xs α (n)) nN est équirépartie modulo 1 si x est irrationnel. Nous prouvons quelques estimations de la discrépance de la suite (xs α (n)) nN .

@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},
     address = {Grenoble},
     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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