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On some properties of Perron numbers (2311.06289v1)
Published 3 Nov 2023 in math.NT
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) $\thetan\ll#D_n(\theta)\ll\sqrt n \thetan.$ (2) $\theta$ is of height $\le \lfloor\theta\rfloor$.