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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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