Optimality of two inequalities for exponents of Diophantine approximation

TL;DR

This paper characterizes when two inequalities involving classical exponents of Diophantine approximation are equalities, providing a complete description of parameter triples and establishing dimension bounds with sharp cases.

Contribution

It offers a full characterization of equality cases for two inequalities of Bugeaud and Laurent in Diophantine approximation and refines the understanding of the joint spectrum of exponent triples.

Findings

Complete description of parameter triples admitting equality.

Lower bounds for Hausdorff and packing dimensions, with sharp cases.

Refined analysis of the joint spectrum of exponent triples.

Abstract

We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit equality for suitable $\ux$, which turns out rather surprising. For $n=2$ our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved $\ux$, and in special cases we can show they are sharp. Proofs are based on the variational principle in parametric geometry of numbers, we enclose sketches of associated combined graphs (templates) where equality is feasible. A twist of our construction provides refined information on the joint spectrum of the respective exponent triples.