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

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)) n∈N est équirépartie modulo 1 si x est irrationnel. Nous prouvons quelques estimations de la discrépance de la suite (xs α (n)) n∈N .

If w=(q k ) k∈N 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) n∈N 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.

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     title = {On the discrepancy of sequences associated with the sum-of-digits function},
     journal = {Annales de l'Institut Fourier},
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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, Tome 37 (1987) no. 3, pp. 1-17. doi : 10.5802/aif.1095. http://www.numdam.org/articles/10.5802/aif.1095/

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