×

Digital sequences with best possible order of \(L_2\)-discrepancy. (English) Zbl 1121.11049

The author of this paper studies the \(L_2\) discrepancy of digital \((0,1)\)-sequences over \(\mathbb{Z}_2\). To be more precise, the paper contains conditions on the generator matrix of a digital \((0,1)\)-sequence which guarantee an \(L_2\) discrepancy of order \(\sqrt{\log N}/N\). A first result given in the article states that the digital \((0,1)\)-sequence generated by a matrix whose \(i\)-th row is made of a string of \(i\) ones followed by infinitely many zeros satisfies \[ NL_{2,N}\ll \sqrt{\log N}, \] where \(L_{2,N}\) is the \(L_2\)-discrepancy of the first \(N\) points of the sequence. Furthermore, the paper gives explicit examples of digital \((0,1)\)-sequences that are not symmetrized sequences, i.e., sequences \((x_n)_{n\geq 0}\) that do not satisfy the relation \(x_{2k+1}=1-x_{2k}\) for \(k\geq 0\), but also satisfy \[ NL_{2,N}\ll \sqrt{\log N}. \]

MSC:

11K38 Irregularities of distribution, discrepancy
11K36 Well-distributed sequences and other variations
11K06 General theory of distribution modulo \(1\)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kuipers, Uniform Distribution of Sequences (1974)
[2] DOI: 10.1007/s006050170054 · Zbl 1108.11309
[3] Haber, J. Res. Nat. Bur. Standards Sect. B70 pp 127– (1966) · Zbl 0158.16002
[4] DOI: 10.1155/S016117129600018X · Zbl 0841.11038
[5] Faure, Acta Arith 55 pp 333– (1990)
[6] Drmota, Sequences, Discrepancies and Applications 1651 (1997) · Zbl 0877.11043
[7] DOI: 10.1007/s00229-005-0577-y · Zbl 1088.11060
[8] Davenport, Mathematika 3 pp 131– (1956)
[9] Chaix, Acta Arith 63 pp 103– (1993)
[10] Zinterhof, Sitzungsber. Österr. Akad. Wiss. Math.-Natur. Kl. II 185 pp 121– (1976)
[11] Roth, Mathematika 1 pp 73– (1959)
[12] DOI: 10.1016/0022-314X(88)90028-5 · Zbl 0654.10050
[13] Proinov, Serdica 11 pp 3– (1985)
[14] DOI: 10.1016/j.jnt.2003.08.002 · Zbl 1048.11061
[15] Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods 63 (1992) · Zbl 0761.65002
[16] DOI: 10.1007/BF01294651 · Zbl 0626.10045
[17] Niederreiter, Diophantine Approximation and its Application pp 129– (1973)
[18] Hellekalek, Acta Arith 67 pp 313– (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.