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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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