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A Remark on the Set of Exactly Approximable Vectors in the Simultaneous Case

Reynold Fregoli

Abstract

We compute the Hausdorff dimension of the set of $ψ$-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than $2$ and for approximating functions $ψ$ with order at infinity less than or equal to $-2$. Our method relies on the analogous result in dimension $1$, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník's Theorem on fibres.

A Remark on the Set of Exactly Approximable Vectors in the Simultaneous Case

Abstract

We compute the Hausdorff dimension of the set of -exactly approximable vectors, in the simultaneous case, in dimension strictly larger than and for approximating functions with order at infinity less than or equal to . Our method relies on the analogous result in dimension , proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník's Theorem on fibres.
Paper Structure (4 sections, 5 theorems, 28 equations)

This paper contains 4 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction and Main Result
  2. Reduction to Fibres
  3. Jarník on Fibres
  4. Proof of Theorem \ref{['prop:mainres']}

Key Result

Theorem 1.1

Assume that the product $q^{2}\cdot\psi(q)$ tends to $0$ as $q$ approaches infinity. Then, one has that where $\lambda$ is the lower order at infinityThe lower order at infinity of a function $g:\mathbb N\to (0,+\infty)$ is defined as \liminf_{x\to \infty}\frac{\log g(x)}{\log x}.The upper order at infinity of the function $g$ is defined analogously but with a limit superior in place of the limit

Theorems & Definitions (8)

  • Theorem 1.1: Bugeaud-Moreira
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof