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Dyadic diaphony of digital nets over \(\mathbb Z_2\). (English) Zbl 1130.11042

Summary: The dyadic diaphony, introduced by P. Hellekalek and H. Leeb [Acta Arith. 58, No. 2, 187–196 (1997; Zbl 0868.11034)], is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over \(\mathbb Z_2\). We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the \(\mathcal L_2\) discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.

MSC:

11K38 Irregularities of distribution, discrepancy
11K06 General theory of distribution modulo \(1\)

Citations:

Zbl 0868.11034
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References:

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