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On the joint distribution of digital sums. (English) Zbl 0678.10037
The author proves a generalization of a theorem of Gel’fond on the distribution of sums of digits: Let m,b\(\geq 2\) be integers, s(n) the sum of digits of n to the base b; \(k_ 1,...,k_{\ell}\) distinct integers with \(b\nmid k_ j\), and \(r_ 1,...,r_{\ell}\) arbitrary integers. If \(k_ jn\equiv r_ j (mod g)\), \(j=1,...,\ell\) has a solution n then \[ \lim_{N\to \infty}(1/N)\quad card\{0\leq n<N:\quad s(k_ jn)\equiv r_ j(m);\quad j=1,...,\ell \}=(\frac{g}{m})^{\ell}\cdot \frac{(d_ 1,...,d_{\ell})}{g}. \] (Here we denote \(d_ j:=(k_ j,g).)\) Otherwise the congruence-system \(s(k_ jn)\equiv r_ j(m)\); \(j=1,...,\ell\) has no solution. This result was shown by Gel’fond for \(\ell =1\). The proof of the multidimensional version is possible by investigating certain exponential sums.
Reviewer: G.Larcher

MSC:
11K06 General theory of distribution modulo \(1\)
11A63 Radix representation; digital problems
11L03 Trigonometric and exponential sums, general
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[1] Davenport, H., ()
[2] Gelfond, A.O., Sur LES nombres qui ont des propriétés additives et multiplicatives données, Acta arithm., 13, 259-265, (1968) · Zbl 0155.09003
[3] Solinas, J., A theorem of metric Diophantine approximation and estimates for sums involving binary digits, ()
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