Dyadic diaphony of digital sequences

Pillichshammer, Friedrich

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

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.

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.

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