ON REAL PARTS OF POWERS OF COMPLEX PISOT NUMBERS

Abstract

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.