Partitions into powers of an algebraic number

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