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arXiv · 1406.0518

Diophantine approximations with Pisot numbers

Victoria Zhuravleva
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Abstract

Let $α$ be a Pisot number. Let $L(α)$ be the largest positive number such that for some $ξ=ξ(α)\in \mathbb R$ the limit points of the sequence of fractional parts $\{ξα^n\}_{n=1}^{\infty}$ all lie in the interval $[L(α), 1-L(α)]$. In this paper we show that if $α$ is of degree at most 4 or $α\le \frac{\sqrt 5 + 1}{2}$ then $L(α)\ge \frac{3}{17}$. Also we find explicitly the value of $L(α)$ for certain Pisot numbers of degree 3.

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Victoria Zhuravleva. 2014-11-14. Diophantine approximations with Pisot numbers. https://arxiv.org/abs/1406.0518

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