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On the discrepancy estimate of normal numbers. (English) Zbl 0947.11023
A number $$\alpha\in (0,1)$$ is said to be normal to base $$q$$ (introduced by E. Borel 1909) if in the $$q$$-ary expansion of $$\alpha$$, $$\alpha= 0.d_1 d_2\dots$$ $$(d_i\in \{0,1,\dots, q-1\})$$ each fixed finite block of digits of length $$k$$ appears with an asymptotic frequency of $$q^{-1}$$ along the sequence $$(d_i)_{i\geq 1}$$. The author constructs a class of normal numbers $$\alpha$$ such that the discrepancy $D(N, (x_n)_{n\geq 1}):= \sup_{x\in (0,1]} \left|\tfrac 1N \#\{0\leq j<N: x_n- [x_n]< x\}-x \right|$ of the sequence $$x_n= \alpha q^n$$ has magnitude $$O(\log^2 N/N)$$.

##### MSC:
 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11K38 Irregularities of distribution, discrepancy 11K06 General theory of distribution modulo $$1$$
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