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A hybrid inequality of Erdös–Turán–Koksma for digital sequences

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Abstract

For bases \(\mathbf{b}=(b_1, \ldots , b_s)\) of \(s\) not necessarily distinct integers \(b_i\ge 2\), we prove a version of the inequality of Erdös–Turán–Koksma for the hybrid function system composed of the Walsh functions in base \(\mathbf{b}^{(1)}=(b_1, \ldots , b_{s_1})\) and, as second component, the \(\mathbf{b}^{(2)}\)-adic functions, \(\mathbf{b}^{(2)}=(b_{s_1+1}, \ldots , b_s)\), with \(s=s_1+s_2\), \(s_1\) and \(s_2\) not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.

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Acknowledgments

The author would like to thank Harald Niederreiter, University of Salzburg, and RICAM, Austrian Academy of Sciences, Linz, for several helpful comments.

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  1. Department of Mathematics, University of Salzburg, Hellbrunnerstrasse 34, 5020 , Salzburg, Austria

    Peter Hellekalek

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  1. Peter Hellekalek
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Correspondence to Peter Hellekalek.

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Communicated by A. Constantin.

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Hellekalek, P. A hybrid inequality of Erdös–Turán–Koksma for digital sequences. Monatsh Math 173, 55–66 (2014). https://doi.org/10.1007/s00605-013-0487-0

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