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Sums of distances to the nearest integer and the discrepancy of digital nets. (English) Zbl 1054.11039

The authors give an explicit formula for the star discrepancy of digital \((0,s,2)\)-nets in base 2 (a class of \(2^s\) points in \([0,1)^2\)). This is achieved by estimating certain sums of distances to the nearest integer and Walsh series analysis. For the special case of the Hammersley point set, where results had earlier been stated by J. H. Halton and S. K. Zaremba [Monatsh. Math. 73, 316–328 (1969; Zbl 0183.31401)], this gives the estimate \(ND_N^*\leq s/3+19/9\) (\(N=2^s\)) improving a bound of H. Niederreiter [Monatsh. Math. 104, 273–337 (1987; Zbl 0626.10045)].
For a special \((0,s,2)\) net in base 2, which is believed to have smallest possible discrepancy, the lower bound \(ND_N^*/\log N\geq 1/5\log 2\) is given answering a question raised by K. Entacher [Monatsh. Math. 130, No. 2, 99–108 (2000; Zbl 0948.11030)].

MSC:

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