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\(L_p\) discrepancy of generalized two-dimensional Hammersley point sets. (English) Zbl 1175.11042

Let \(b\geq 2\) be an integer and let \(\Sigma=(\sigma_r)_{r\geq 0}\) be a sequence of permutations of \(\{0,1,\dots, b-1\}\). For any integers \(n\) and \(N\) with \(n\geq 0\) and \(1\leq N\leq b^n\), write \(N-1= \sum^\infty_{r=0} a_r(N)b^r\) in the \(b\)-adic system, where \(a_r(N)= 0\) if \(r\geq n\). The authors define the generalized van der Corput sequence \(S^\Sigma_b\) in base \(b\) associated to \(\Sigma\) by \[ S^\Sigma_b(N)= \sum^\infty_{r=0} {\sigma_r(a_r(N))\over b^{r+1}}\quad\text{for all }N\geq 1, \] and the generalized two-dimensional Hammersley point set in base \(b\) consisting of \(b^n\) points associated to \(\Sigma\) by \[ {\mathcal H}^\Sigma_{b,n}= \Biggl\{\Biggl(S^\Sigma_b(N),\,{N-1\over b^n}\Biggr)\,(1\leq N\leq b^n)\Biggr\}, \] respectively. In particular, they consider the special case \(\Sigma= (\sigma_0,\dots,\sigma_{n- 1},id,id,\dots)\), where \(id\) is the identical permutation. They give the explicit formulas of \(L_1\) and \(L_2\) discrepancy of the classical two-dimensional Hammersley point set in. In arbitrary base \(b\geq 2\) consisting of \(b^n\) points, generalizing the previous results (in base \(b=2\)) obtained by the second author [Monatsh. Math. 136, No. 1, 67–79 (2002; Zbl 1010.11043)] and others, and show that the \(L_p\) \((p\in\mathbb N)\) discrepancy of this point set is not of best possible order with respect to the general results of Roth and Schmidt. They also prove that there always exist sequences \(\Sigma\) such that the \(L_p\) discrepancy of the generalized two-dimensional Hammersley point set is of best possible order. Moreover, for such \(\Sigma\) the formula of \(L_2\) discrepancy is given explicitly.

MSC:

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

Citations:

Zbl 1010.11043
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References:

[1] Beck J., Chen W.W.L.: Irregularities of Distribution. Cambridge University Press, London (1987) · Zbl 0617.10039
[2] Chaix H., Faure H.: Discrépance et diaphonie en dimension un. Acta Arith. 63, 103–141 (1993) · Zbl 0772.11022
[3] De Clerck L.: A method for the exact calculation of the star-discrepancy of plane sets applied to the sequences of Hammersley. Monatsh. Math. 101, 261–278 (1986) · Zbl 0588.10059
[4] Drmota M., Tichy R.F.: Sequences, discrepancies and applications. In: Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997) · Zbl 0877.11043
[5] Faure H.: Discrépance de suites associées à un système de numération (en dimension un). Bull. Soc. Math. Fr. 109, 143–182 (1981) · Zbl 0488.10052
[6] Faure H.: On the star-discrepancy of generalized Hammersley sequences in two dimensions. Monatsh. Math. 101, 291–300 (1986) · Zbl 0588.10060
[7] Faure H.: Discrepancy and diaphony of digital (0,1)-sequences in prime bases. Acta Arith. 117, 125–148 (2005) · Zbl 1080.11054
[8] Faure H.: Improvements on low discrepancy one-dimensional sequences and two-dimensional point sets. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006., pp. 327–341. Springer, Berlin (2008) · Zbl 1213.11153
[9] Halton J.H., Zaremba S.K.: The extreme and the L 2 discrepancies of some plane sets. Monatsh. Math. 73, 316–328 (1969) · Zbl 0183.31401
[10] Kritzer P.: On some remarkable properties of the two-dimensional Hammersley point set in base 2. J. Théor. Nombres Bordeaux 18, 203–221 (2006) · Zbl 1103.11024
[11] Kritzer P., Larcher G., Pillichshammer F.: A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets. Ann. Mat. Pura Appl. 186, 229–250 (2007) · Zbl 1150.11026
[12] Kritzer P., Pillichshammer F.: Improvements of the discrepancy of the van der Corput sequence. Math. Pannonica 16, 179–198 (2005) · Zbl 1119.11045
[13] Kritzer P., Pillichshammer F.: An exact formula for the L 2 discrepancy of the shifted Hammersley point set. Uniform Distrib. Theory 1, 1–13 (2006) · Zbl 1147.11041
[14] Kritzer P., Pillichshammer F.: Point sets with low L p -discrepancy. Math. Slovaca 57, 11–32 (2007) · Zbl 1153.11037
[15] Kuipers L., Niederreiter H.: Uniform Distribution of Sequences. Wiley, New York (1974) · Zbl 0281.10001
[16] Matoušek J.: Geometric Discrepancy. Algorithms and Combinatorics, vol. 18. Springer, Berlin (1999)
[17] Pillichshammer F.: On the L p -discrepancy of the Hammersley point set. Monatsh. Math. 136, 67–79 (2002) · Zbl 1010.11043
[18] Roth K.F.: On irregularities of distribution. Mathematika 1, 73–79 (1954) · Zbl 0057.28604
[19] Schmidt, W.M.: Irregularities of distribution X. In: Number Theory and Algebra, pp. 311–329. Academic Press, New York (1977) · Zbl 0373.10020
[20] Strauch O., Porubský Š.: Distribution of Sequences: A Sampler. Peter Lang, Bern (2005)
[21] Vilenkin, I.V.: Plane nets of integration. Ž. Vyčisl. Mat. i Mat. Fiz. 7, 189–196 (1967) [English translation in: U.S.S.R. Comput. Math. Math. Phys. 7(1), 258–267 (1967)] · Zbl 0187.10701
[22] White B.E.: Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix. Monatsh. Math. 80, 219–229 (1975) · Zbl 0322.65002
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