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# Kriterien in der Theorie der Gleichverteilung

Criteria in the theory of uniform distribution

## Abstract

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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## Literatur

1. [1]

Hlawka, E.: Zur quantitativen Theorie der Gleichverteilung. Sitzber. Österr. Akad. Wiss., Math.-naturw. Kl., Abt. II184, 355–365 (1975).

2. [2]

Kuipers, L., andH. Niederreiter: Uniform Distribution of Sequences. New York: Wiley Interscience. 1974.

3. [3]

Niederreiter, H.: Almost-arithmetic progressions and uniform distribution. Trans. Amer. Math. Soc.161, 283–292 (1971).

4. [4]

Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill Book Company. 1974.

5. [5]

Rindler, H.: ZurL 1-Gleichverteilung auf Abelschen und kompakten Gruppen. Arch. Math.26, 209–213 (1975).

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