Approximation to an extremal number, its square and its cube

TL;DR

This paper determines the approximation constants for the cube of an extremal number, extending known results from lower powers and providing insights into the approximation properties of these special numbers.

Contribution

It explicitly computes all classic approximation constants for the cube of an extremal number, a case previously unexplored, and links Khintchine's inequalities with uniform approximation.

Findings

Exact values of approximation constants for n=3

Detailed analysis of the combined graph for extremal numbers and their powers

Connections established between Khintchine's inequalities and uniform approximation

Abstract

We study rational approximation properties for successive powers of extremal numbers defined by Roy. For $n\in{\{1,2\}}$, the classic approximation constants $\lambda_{n}(\zeta),\hat{\lambda}_{n}(\zeta),w_{n}(\zeta),\hat{w}_{n}(\zeta)$ connected to an extremal number $\zeta$ have been established and in fact much more is known. However, so far almost nothing had been known for $n\geq 3$. In this paper we determine all classic approximation constants as above for $n=3$. Our methods will more generally provide detailed information on the combined graph defined by Schmidt and Summerer assigned to an extremal number, its square and its cube. We provide some results for $n=4$ as well. In the course of the proof of the main results we establish a very general connection between Khintchine's transference inequalities and uniform approximation.