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On a method of constructing well-dispersed sequences. (English. Russian original) Zbl 0876.11039
Discrete Math. Appl. 5, No. 4, 381-392 (1995); translation from Diskretn. Mat. 7, No. 3, 89-99 (1995).
The author constructs uniformly distributed sequences on a rather general class of domains including multidimensional cubes. Similar in its basic idea to the classical van der Corput sequence [see L. Kuipers and H. Niederreiter, Uniform distribution of sequences (1974; Zbl 0281.10001) for details], one uses a so-called “radix-$$Q$$” representation of natural numbers to construct well-dispersed sequences. Here $$Q$$ denotes a sequence of natural numbers $$q_m\geq 2$$. In the case of the constant sequence $$q_m=q$$, this will yield the ordinary $$q$$-adic representation.
The author uses a method of constructing infinite trees with levels assigned to their vertices and numbers assigned to their edges to introduce this representation. For a given probability space $$(\Omega, {\mathcal B},\mu)$$ and a sequence of mappings $$f_j$$ of multiplicity $$q_j\geq 2$$ from $$\Omega$$ to itself, he defines a sequence of points $$\xi^{(m)}: =f_m (\xi^{(m-1)})$$ in $$\Omega$$. He also analyzes how many operations will be needed to construct the point $$\xi_m$$ and how the variance $$\rho_N: =\sup_{\eta \in\Omega} \min_{n=0, \dots, N-1}|\eta-\xi_n |$$ behaves (where we assume $$\Omega$$ to support a metric). A further result concerns an estimate on the discrepancy taken with respect to sets with simple boundaries. The construction method presented in this paper extends similar approaches of A. Lambert [Proc. Am. Math. Soc. 103, 383-388 (1988; Zbl 0655.10055)] and P. Hellekalek [J. Number Theory 18, 41-55 (1984; Zbl 0531.10055)], which are based on Cantor series expansions.
##### MSC:
 11K06 General theory of distribution modulo $$1$$ 11K31 Special sequences
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