Optimality of two inequalities for exponents of Diophantine approximation

Abstract

We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to ξ_∈Rn. We provide a complete description of parameter triples that admit equality for suitable ξ_, which turns out rather surprising. We implicitly also obtain partial results on the optimality of another related inequality of Schmidt and Summerer. For n=2 our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved ξ_, 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.