Sets of Exact(er) approximation order

In this talk we introduce a quantitative notion of exactness within Diophantine approximation. Given functions Ψ : (0, ∞) → (0, ∞) and ω : (0, ∞) → (0, 1), we study the set of points that are Ψ-well approximable but not Ψ(1 − ω)-well approximable, denoted E(Ψ,ω). This generalises the set of Ψ-exact approximation order as studied by Bugeaud (Math. Ann. 2003). We prove results on the cardinality and Hausdorff dimension of E(Ψ,ω). In particular, for certain functions Ψ we find a critical threshold on ω whereby the set E(Ψ,ω) drops from positive Hausdorff dimension to empty when ω is multiplied by a constant. The results discussed can be found in [2510.18451] A quantitative framework for sets of exact approximation order by rational numbers.