Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

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Sets of exact approximation order by rational numbers (2003, background): “Jarńık’s result is, however, not strong enough to imply that Exact(Ψ) and K(Ψ) have the same Hausdorff dimension, a problem raised by Beresnevich, Dickinson and Velani at the end of [1].”
Diophantine approximation and Cantor sets (2008, methodology): “This result is new and cannot be obtained via the metric theory as used in [11], see [3] and the Remark at the end of Sect.”
On a problem of K. Mahler: Diophantine approximation and Cantor sets (2005, background): “Remark. It is evident from the above proof that the statement of Corollary 2 does not in anyway utilize the full power of Theorem 1. The argument outlined above can be modified to establish much stronger ‘exact order’ statements in the spirit of those in [ 2 ].”
Exponents of Diophantine approximation (2006, background): “et m,nbe integers with 1 ≤ n≤ m. Does the inequality λm(ξ) ≥ nλn(ξ)−m+ n m hold for every transcendental real number ξ? The next problem extends a question posed by Beresnevich, Dickinson, and Velani [9] in the case of (simultaneous) rational approximation. Problem 9.11. Let nbe a positive integer a”