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Discrepancy and diaphony in one dimension. (DiscrĂ©pance et diaphonie en dimension un.) (French) Zbl 0772.11022

This paper contains several interesting results concerning the \(L^ p\)- discrepancy \(D^{(p)}(N,X)\) and the diaphony \(F(N,X)\) of special point sequences \(X=(x_ n)\) in the unit interval \([0,1]\). The authors mainly consider generalized van der Corput sequences \(S_ b^ \Sigma\) in base \(b\) associated to a sequence \(\Sigma=(\sigma_ j)_{j\geq 0}\) of permutations of \(\{0,\dots,b-1\}\), defined by \(S_ b^ \Sigma(n)=\sum_{j=0}^ \infty \sigma_ j(a_ j(n))b^{-j-1}\), where \(a_ j(n)\) is the \(j\)th digit of \(n-1\) in base \(b\). The main results are explicit formulae for the \(L^ 2\)-discrepancies and the diaphonies of such sequences as well as generalizations to variable bases.
Reviewer: R.F.Tichy (Graz)

MSC:

11K06 General theory of distribution modulo \(1\)
11K31 Special sequences
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