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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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##### References:
 [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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