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Kriterien in der Theorie der Gleichverteilung

Criteria in the theory of uniform distribution


It is the aim of this paper to introduce two new notions of discrepancy. They are defined by the formulas

$$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z e^2 \pi i\omega \left( n \right)} \right)} - f\left( 0 \right)} \right|, and \hfill \\ \delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z \omega \left( n \right)} \right)} \cdot z - \int\limits_0^z {f\left( \zeta \right)d\zeta } } \right|, \hfill \\ \end{gathered} $$

wheref is a holomorphic function defined in the unit disc withf (k)(0)≠0 for allk∈ℕ,r<1 is a positive number, and ω is a sequence in [0, 1]. The first of these discrepancies can be generalized for multidimensional sequences. ω is uniform distributed if and only if lim N→∞ Δ N r (ω;f)=0 resp. lim N→∞δ N r (ω;f)=0. These results are proved in a quantitative way by estimating the classical discrepancyD N (ω) by means ofΔ N r (ω;f) and δ N r (ω;f):

$$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Phi \left( {\Delta _N^r \left( {\omega ;f} \right)} \right), \hfill \\ \delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Psi \left( {\delta _N^r \left( {\omega ;f} \right)} \right). \hfill \\ \end{gathered} $$

The functions Φ and Ψ only depend onf andr. These estimations are based on the inequalities ofKoksma-Hlawka andErdös-Turán.

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Taschner, R.J. Kriterien in der Theorie der Gleichverteilung. Monatshefte für Mathematik 86, 221–237 (1978).

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