Let ψ:R+→R+ψ:R+→R+ be a non-increasing function. A real number xx is said to be ψψ-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system
|qx−p|<ψ(t) and |q|<t|qx−p|<ψ(t) and |q|<t
has a non-trivial integer solution for all large enough tt. Denote the collection of such points by D(ψ)D(ψ). In this talk, I will explain metrical theory (Lebesgue measure and Hausdorff measure/dimensions) associated with the complement D(ψ)cD(ψ)c (the set of ψψ-Dirichlet non-improvable numbers).
This is a joint work with D. Kleinbock, N. Wadleigh and B-W. Wang.