\(L_p\) 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.


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


Zbl 1010.11043
Full Text: DOI


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