On the discrepancy estimate of normal numbers. (English) Zbl 0947.11023

Levin, M. B.

Acta Arith. 88, No. 2, 99-111 (1999).

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 \(O(\log^2 N/N)\).

Reviewer: L.Heinrich (Augsburg)

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 \(1\)

Keywords:

Korobov’s problem; van der Corput-type sequences; normal numbers; discrepancy