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The mixed Littlewood conjecture for pseudo-absolute values

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Abstract

In this paper we study the mixed Littlewood Conjecture with pseudo-absolute values. We show that if \(p\) is a prime and \(\mathcal D \) is a pseudo-absolute value sequence satisfying mild conditions then

$$\begin{aligned} \inf _{n\in \mathbb N } n|n|_p|n|_\mathcal D \Vert n\alpha \Vert =0\quad \text{ for } \text{ all }\,\,\alpha \in \mathbb R . \end{aligned}$$

Our proof relies on a measure rigidity theorem due to Lindenstrauss and lower bounds for linear forms in logarithms due to Baker and Wüstholz. We also deduce the answer to the related metric question of how fast the infimum above tends to zero, for almost every \(\alpha \).

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Acknowledgments

We would like to thank Sanju Velani for encouraging us to look at these problems, which seem to have first been proposed in a systematic way in [1, Section 1.3]. The second author would like to thank Barak Weiss for helpful conversations regarding this project and the first author would also like to thank Simon Kristensen for various discussions. We would also like to thank the referee, whose suggestions resulted in the separation of Theorem 2 from the proof of Theorem 1, among other improvements.

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Authors and Affiliations

  1. Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 , Aarhus C, Denmark

    Stephen Harrap

  2. Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Alan Haynes

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  1. Stephen Harrap
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  2. Alan Haynes
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Correspondence to Alan Haynes.

Additional information

AH: Research supported by EPSRC grant EP/F027028/1, and by a fellowship from the Heilbronn Institute for Mathematical Research.

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Harrap, S., Haynes, A. The mixed Littlewood conjecture for pseudo-absolute values. Math. Ann. 357, 941–960 (2013). https://doi.org/10.1007/s00208-013-0928-z

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  • DOI: https://doi.org/10.1007/s00208-013-0928-z

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