The sum-of-digits-function and uniform distribution modulo 1. (English) Zbl 0990.11053
Let \(s_q(n)\) be the \(q\)-ary sum-of-digits-function for a fixed integer \(q>1\). M. Mendès France [J. Number Theory 5, 1-15 (1973; Zbl 0252.10033)] and J. Coquet [Acta Arith. 36, 157-162 (1980; Zbl 0429.10032)] showed that the sequence \((\{\alpha\cdot s_q(n)\})\) is uniformly distributed modulo \(1\) for irrational \(\alpha\). R. F. Tichy and G. Turnwald [J. Number Theory 26, 68-78 (1987; Zbl 0628.10052)] established estimates for the discrepancy of \((\{\alpha\cdot s_q(n)\})\) for irrational \(\alpha\).
In the first part of this paper refined estimates for the discrepancy of \((\{\alpha\cdot s_q(n)\})\) is given. In the second part a first result concerning the uniform distribution of \(d\)-dimensional sequences \((\{\alpha_1\cdot s_{q_1}(n)\}, \dots, \{\alpha_d\cdot s_{q_d}(n)\})\) is proved, where \(\alpha_1\), \(\dots\), \(\alpha_d\) are fixed irrational numbers and \(q_1,\dots,q_d>1\) are fixed pairwise coprime integers.
In the first part of this paper refined estimates for the discrepancy of \((\{\alpha\cdot s_q(n)\})\) is given. In the second part a first result concerning the uniform distribution of \(d\)-dimensional sequences \((\{\alpha_1\cdot s_{q_1}(n)\}, \dots, \{\alpha_d\cdot s_{q_d}(n)\})\) is proved, where \(\alpha_1\), \(\dots\), \(\alpha_d\) are fixed irrational numbers and \(q_1,\dots,q_d>1\) are fixed pairwise coprime integers.
Reviewer: Takao Komatsu (Tsu)
MSC:
| 11K06 | General theory of distribution modulo \(1\) |
| 11A63 | Radix representation; digital problems |
| 11K38 | Irregularities of distribution, discrepancy |
| 11K16 | Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. |
References:
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