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A good permutation for one-dimensional diaphony. (English) Zbl 1206.11096

Summary: We focus on two aspects of one-dimensional diaphony \(F(S^\sigma_b,N)\) of generalised van der Corput sequences in arbitrary bases. First we give a permutation with the best distribution behaviour concerning the diaphony known so far. We improve a result of Chaix and Faure from 1993 from a value of 1.31574\(\dots\) for a permutation in base 19 to 1.13794 \(\dots\) for our permutation in base 57. Moreover for an infinite sequence \(X\) and its symmetric version \(\tilde X\), we analyse the connection between the diaphony \(F(X, N)\) and the \(L_{2}\)-discrepancy \(L_2 (\tilde X,N)\) using another result of Chaix and Faure. Therefore we state an idea how to get a lower bound for the diaphony of generalised van der Corput sequences in arbitrary base \(b\).

MSC:

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

[1] Faure H., Bull. Soc. Math. France 109 pp 143– (1981)
[2] Faure H., Acta Arith. 55 pp 333– (1990)
[3] DOI: 10.1016/0022-314X(92)90107-Z · Zbl 0768.11026 · doi:10.1016/0022-314X(92)90107-Z
[4] Chaix H., Acta Arith. 63 pp 103– (1993)
[5] DOI: 10.4064/aa117-2-2 · Zbl 1080.11054 · doi:10.4064/aa117-2-2
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