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Möbius orthogonality of sequences with maximal entropy

Abstract

We prove that strongly b-multiplicative functions of modulus 1 along squares are asymptotically orthogonal to the Möbius function. This provides examples of sequences having maximal entropy and satisfying this property.

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Acknowledgements

The authors wish to thank Mariusz Lemańczyk for pointing out valuable references to the literature concerning Möbius orthogonality of positive entropy dynamical systems, and helping us place our result into the context of related research works. Moreover, we thank Clemens Müllner for several fruitful discussions.

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Correspondence to Lukas Spiegelhofer.

Additional information

This work was supported by the joint ANR-FWF-project ANR-14-CE34-0009, I-1751 MuDeRa.

The first and fourth authors are supported by the Austrian Science Foundation FWF, SFB F55-02 “Subsequences of Automatic Sequences and Uniform Distribution”.

Christian Mauduit has sadly passed away before the publication of this paper.

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Drmota, M., Mauduit, C., Rivat, J. et al. Möbius orthogonality of sequences with maximal entropy. JAMA 146, 531–548 (2022). https://doi.org/10.1007/s11854-022-0201-z

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  • DOI: https://doi.org/10.1007/s11854-022-0201-z