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Fractional parts of powers of negative rationals

Qing Lu, Weizhe Zheng

Abstract

We prove that for any real number $ξ\neq 0$ and any coprime integers $p>q\ge1$ such that $ξ$ is irrational or $q>1$, the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-p/q)^n)_{n\ge 0}$ is not contained in any interval of length less than $(1+q/p-q^2/p^2)/p$.

Fractional parts of powers of negative rationals

Abstract

We prove that for any real number and any coprime integers such that is irrational or , the image in of the sequence is not contained in any interval of length less than .
Paper Structure (3 sections, 6 theorems, 29 equations)

This paper contains 3 sections, 6 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:2']}
  3. Proof of Theorem \ref{['t:1']}

Key Result

Theorem 1.1

Let $\xi\neq 0$ and $\eta$ be real numbers and let $p>q\ge 1$ be integers. Assume that $\xi\notin\mathbb{Q}$ or $p/q\notin \mathbb{Z}$. Then where $r=q/p$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['t:2']}
  • Lemma 3.1
  • ...and 2 more