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Partitions into powers of an algebraic number

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Abstract

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number \(\beta \). We prove that if \( \beta \) is real quadratic, then the number of partitions is always finite if and only if some conjugate of \(\beta \) is larger than 1. Further, we show that for \(\beta \) satisfying a certain condition, the partition function attains all non-negative integers as values.

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Acknowledgements

We thank Jakub Krásenský, Zuzana Masáková, and Edita Pelantová for helpful discussions, and to Shigeki Akiyama and Artūras Dubickas for suggestions of several useful references. Finally, we thank the anonymous referees for important suggestions and corrections, especially for a simplification of Lemma 6 and its proof.

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  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Vítězslav Kala & Mikuláš Zindulka

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  1. Vítězslav Kala
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Correspondence to Vítězslav Kala.

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The authors were supported by Czech Science Foundation (GAČR) grant 21-00420M, Charles University Research Centre program UNCE/SCI/022, and GA UK No. 742120.

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Kala, V., Zindulka, M. Partitions into powers of an algebraic number. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00845-2

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