Simultaneous rational approximation to successive powers of a real number

TL;DR

This paper introduces new tools to improve bounds on the approximation of powers of transcendental numbers and refines the lower bounds for algebraic integer approximations, advancing understanding in Diophantine approximation.

Contribution

The authors develop novel methods that significantly improve upper bounds for the uniform exponent of rational approximation to powers of a transcendental number.

Findings

Improved upper bounds for $\widehat{\lambda}_n(\xi)$ for all $n \\ge 4$

Refined lower bounds for approximation exponents by algebraic integers

Quantitative enhancement over previous bounds with explicit constants

Abstract

We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehat{\lambda}_n(\xi)$ to successive powers $1,\xi,\dots,\xi^n$ of a given real transcendental number $\xi$. As an application, we obtain a refined lower bound for the exponent of approximation to $\xi$ by algebraic integers of degree at most $n+1$. The new lower bound is $n/2+a\sqrt{n}+4/3$ with $a=(1-\log(2))/2\simeq 0.153$, instead of the current $n/2+\mathcal{O}(1)$.