Some examples of application of the metric entropy method

Abstract

We show how the metric entropy method can be substituted for the dyadic chaining, to prove in a unified setting several classical results. Among them are Stechkin's theorem, Gal--Koksma theorems and quantitative Borel--Cantelli lemmas. We give simpler proofs and improve some of these results. Two classes of examples are given: firstly stationary Gaussian sequences with applications to upper and lower classes and the law of the iterated logarithm for subsequences, and secondly in Diophantine approximation relatively to Gal and Schmidt's theorems.