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.

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.

Reviewer: P.Hellekalek (Salzburg)