Diophantine approximations with Pisot numbers

TL;DR

This paper investigates the behavior of fractional parts of sequences involving Pisot numbers, establishing lower bounds for a specific limit point interval and explicitly computing these bounds for certain degree 3 Pisot numbers.

Contribution

It provides new lower bounds for the limit point interval of fractional parts for Pisot numbers of degree up to 4 and certain degree 3 cases, advancing understanding of Diophantine approximations.

Findings

For degree ≤ 4 or α ≤ (√5+1)/2, L(α) ≥ 3/17.

Explicit values of L(α) are computed for specific degree 3 Pisot numbers.

The results improve bounds on fractional parts for a class of Pisot numbers.

Abstract

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