Dyadic diaphony of digital sequences

Pillichshammer, Friedrich

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 501-521.

La diaphonie diadique est une mesure quantitative pour l’irrégularité de la distribution d’une suite dans le cube unitaire. Dans cet article nous donnons des formules pour la diaphonie diadique des $(0,s)$-suites digitales sur ${\mathbb{Z}}_2$, $s=1,2$. Ces formules montrent que, pour $s\in \{1,2\}$ fixé, la diaphonie diadique a les mêmes valeurs pour chaque $(0,s)$-suite digitale. Pour $s=1$, il résulte que la diaphonie diadique et la diaphonie des $(0,1)$-suites digitales particulières sont égales, en faisant abstraction d’une constante. On détermine l’ordre asymptotique exact de la diaphonie diadique des $(0,s)$-suites digitales et on montre que pour $s=1$ elle satisfait un théorème de la limite centrale.

The dyadic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we give formulae for the dyadic diaphony of digital $(0,s)$-sequences over ${\mathbb{Z}}_2$, $s=1,2$. These formulae show that for fixed $s\in \{1,2\}$, the dyadic diaphony has the same values for any digital $(0,s)$-sequence. For $s=1$, it follows that the dyadic diaphony and the diaphony of special digital $(0,1)$-sequences are up to a constant the same. We give the exact asymptotic order of the dyadic diaphony of digital $(0,s)$-sequences and show that for $s=1$ it satisfies a central limit theorem.

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