## Good permutations for extreme discrepancy.(English)Zbl 0768.11026

It is known that for every infinite sequence $$X: x_ 1,x_ 2,\dots$$ in $$[0,1)$$ for the discrepancy $$D_ N$$ of the first $$N$$ sequence elements we have $$s(x):= \limsup_ n(D(N)/\text{Log }N) \geq 0.12$$. For the usual van der Corput-sequence to base $$b$$, the value $$s(x)$$ tends to infinity with growing base $$b$$.
In this paper it is shown that in the case of generalized van der Corput- sequences for every base $$b$$ there exists a permutation $$\sigma$$ such that for the generalized van der Corput-sequence $$S^ \sigma_ b$$ one has $$S(S^ \sigma_ b) \leq 1/\text{Log }2$$. As a special case for base $$b = 36$$ a permutation $$\sigma$$ is explicitly given such that $$S(S^ \sigma_{36}) = 23/(35\text{ Log }6) = 0.3667\dots$$. This is the smallest value for $$s$$ known till now.

### MSC:

 11K38 Irregularities of distribution, discrepancy
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### References:

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