## $$L_p$$ 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

### Keywords:

$$L_p$$ discrepancy; Hammersley point set

Zbl 1010.11043
Full Text:

### References:

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