Approximation to an extremal number, its square and its cube

Johannes Schleischitz

Vol. 287 (2017), No. 2, 485–510

DOI: 10.2140/pjm.2017.287.485

Abstract

We study rational approximation properties for successive powers of extremal numbers defined by Roy. For $n\in \left\{1,2\right\}$, the classic approximation constants ${\lambda }_n\left(\zeta \right),{\widehat{\lambda }}_n\left(\zeta \right),w_n\left(\zeta \right),{\hat{w}}_n\left(\zeta \right)$ 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 $n\ge 3$. 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 proofs of the main results we establish a very general connection between Khintchine’s transference inequalities and uniform approximation.

Keywords

extremal numbers, Diophantine approximation constants, geometry of numbers, lattices

Mathematical Subject Classification 2010

Primary: 11J13

Secondary: 11H06

Milestones

Received: 9 March 2016

Revised: 22 August 2016

Accepted: 26 September 2016

Published: 9 March 2017