On some properties of Perron numbers

Abstract

Let $\theta$ be a real number, $n\in\mathbb N$, and $D_n(\theta)=\left\{\sum_{k=1}^n a_k\theta^{k}\mid a_k\in\{0,\dots,\lfloor \theta\rfloor\}\right\}. $ Let $\theta$ be a Perron number, that is, an algebraic integer $>1$ whose other Galois conjugates are less than $\theta$ in absolute value. I shall prove two results: (1) $\theta^n\ll\#D_n(\theta)\ll\sqrt n \theta^n.$ (2) $\theta$ is of height $\le \lfloor\theta\rfloor$.