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Dyadic diaphony of digital sequences. (English) Zbl 1152.11036

The author considers the dyadic diaphony \(F_{2,N}(\omega)\) [see P. Hellekalek, H. Leeb, Acta Arith. 80, 2, 187–196 (1997; Zbl 0868.11034)] of digital \((0,s)\)-sequences \(\omega\) with \(s\in\{1,2\}\). For \((0,1)\)-sequences \(\omega\) he proves \[ (N\,F_{2,N}(\omega))^2=3\sum_{u=1}^\infty \left\| \frac{N}{2^u}\right\| ^2 \] and derives some interesting consequences. In particular he shows that the dyadic diaphony of \((0,1)\) sequences satisfies a central limit theorem. Similar for \((0,2)\)-sequences \(\omega\) the author obtains \[ (N\,F_{2,N}(\omega))^2=\frac 94\sum_{u=1}^\infty u\left\| \frac{N}{2^u}\right\| ^2. \] in this case the author cannot prove a central limit theorem, but a similar weaker theorem.

MSC:

11K06 General theory of distribution modulo \(1\)
11A63 Radix representation; digital problems
11K38 Irregularities of distribution, discrepancy
65C05 Monte Carlo methods
65C20 Probabilistic models, generic numerical methods in probability and statistics

Citations:

Zbl 0868.11034
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References:

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