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

TL;DR

This paper revisits bounds on uniform rational approximation exponents for quadruples of real numbers in geometric progression, explaining why previous bounds are not optimal.

Contribution

It provides an analysis showing that the previously established upper bounds for approximation exponents can be improved.

Findings

Previous bounds are not optimal for approximation exponents.

The paper offers insights into the limitations of earlier bounds.

It suggests directions for tighter bounds in future work.

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.