Diophantine sets and Dirichlet improvability

摘要

This note pushes further the discussion by Beresnevich, Guan, Marnat, Ramirez, and Velani (href{https://doi.org/10.1016/j.aim.2022.108316}{{em Adv.~Math./} {bf 401} (2022), art.~id.~108316}) about relations between Dirichlet improvable, badly approximable and singular points by considering Diophantine sets extending the notion of bad approximability.