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Regularities in the distribution of special sequences
Authors:Peter Hellekalek
Institution:1. U.E.R. de Mathématiques, Université de Provence, 3, Place Victor Hugo, Marseille, France;2. Institut für Mathematik, Universität in Salzburg, Petersbrunnstrasse 19, A-5020 Salzburg, Austria
Abstract:For a given sequence ω = (x(k))k=0 in Us, U = R/Z, let S(ω) denote the set of all boxes I in Us with bounded discrepancy function, i.e.,
/></figure> l<sub><em>I</em></sub> the characteristic function of <em>I</em>. In this paper the set <em>S</em>(<em>ω</em>) is studied for a type of sequences generalizing the Halton-sequences. <em>S</em>(<em>ω</em>) is completely determined in the one-dimensional case. In higher dimensions the subset <figure class=/></figure> of <em>S</em>(<em>ω</em>) is determined and a necessary condition on the elements of <em>S</em>(<em>ω</em>) is proved. The methods used in the proofs belong to ergodic theory.</td>
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