Partitions into powers of an algebraic number

Mikul\'a\v{s} Zindulka; V\'it\v{e}zslav Kala

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Partitions into powers of an algebraic number

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arxiv 2305.16688 v2 pith:NL4J3M2L submitted 2023-05-26 math.NT

Partitions into powers of an algebraic number

V\'it\v{e}zslav Kala , Mikul\'a\v{s} Zindulka This is my paper

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keywords betanumberpartitionsalgebraicnon-negativepowersalwaysattains

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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.

Partitions into powers of an algebraic number

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