## Discrepancy and diaphony in one dimension. (Discrépance 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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