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A general discrepancy estimate based on $$p$$-adic arithmetics. (English) Zbl 1223.11097
The classical Erdös-Turan inequality allows to estimate the discrepancy of a finite point set by a certain sum of evaluations of trigonometric functions. In the paper under review, the author generalizes this inequality to a new class of functions, which arises in a natural way from the dual group of the $$p$$-adic integers $$\mathbb{Z}_p$$ ($$p$$ a prime). Using the same function system, the author also presents a new form of Weyl’s criterion for uniform distribution (modulo 1) of infinite sequences. The results in the paper are particularly easy to apply to sequences which are constructed using the $$p$$-adic radical inverse function, such as the van der Corput sequence.

##### MSC:
 11K38 Irregularities of distribution, discrepancy 11K06 General theory of distribution modulo $$1$$ 11K41 Continuous, $$p$$-adic and abstract analogues 11K45 Pseudo-random numbers; Monte Carlo methods 11L03 Trigonometric and exponential sums (general theory)
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