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On simultaneous approximation to a real number, its square, and its cube, II

Damien Roy

TL;DR

The paper investigates the exponent of uniform rational approximation to the geometric progression $(1,\xi,\xi^2,\xi^3)$ and shows that the previously proposed upper bound $\lambda_3\approx 0.4245$ is not optimal. It introduces polynomial maps $C$ and $E$ and the auxiliary maps $\Psi_\pm$ to study minimal points and derive a rigid algebraic structure among consecutive minimal points when $\lambda=\lambda_3$. Through a sequence of steps, including two new sets of algebraic relations and a final contradiction, the author proves $\lambda<\lambda_3$, contributing toward determining the least upper bound for $\widehat{\lambda}_3(\xi)$. The addendum highlights an additional relation via a map $\Xi$ that could guide future improvements and candidate constructions for extremal numbers. Overall, the work blends Diophantine approximation, lattice geometry, and multilinear polynomial maps to refine bounds on simultaneous rational approximation in cubic order.

Abstract

In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper bound is not optimal.

On simultaneous approximation to a real number, its square, and its cube, II

TL;DR

The paper investigates the exponent of uniform rational approximation to the geometric progression and shows that the previously proposed upper bound is not optimal. It introduces polynomial maps and and the auxiliary maps to study minimal points and derive a rigid algebraic structure among consecutive minimal points when . Through a sequence of steps, including two new sets of algebraic relations and a final contradiction, the author proves , contributing toward determining the least upper bound for . The addendum highlights an additional relation via a map that could guide future improvements and candidate constructions for extremal numbers. Overall, the work blends Diophantine approximation, lattice geometry, and multilinear polynomial maps to refine bounds on simultaneous rational approximation in cubic order.

Abstract

In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of -linearly independent real numbers in geometric progression. Here, we explain why this upper bound is not optimal.
Paper Structure (9 sections, 38 theorems, 167 equations)

This paper contains 9 sections, 38 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The maps $C$ and $E$
  4. The maps $\Psi_-$ and $\Psi_+$
  5. First step
  6. A new set of algebraic relations
  7. Another set of algebraic relations
  8. Final contradiction
  9. Addendum

Key Result

Theorem 1.1

Let $\xi\in\mathbb{R}$ with $[\mathbb{Q}(\xi)\colon\mathbb{Q}] >3$, and let $c$ and $\lambda$ be positive real numbers. Suppose that, for any sufficiently large value of $X$, the inequalities admit a non-zero solution $\mathbf{x}=(x_0,x_1,x_2,x_3)\in \mathbb{Z}^4$. Then, we have $\lambda\le \lambda_3$. Moreover, if $\lambda=\lambda_3$, then $c$ is bounded below by a positive constant depending on

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • ...and 59 more