Diophantine sets and Dirichlet improvability

TL;DR

This paper explores the relationships between various classes of points in Diophantine approximation, focusing on Diophantine sets that extend badly approximable points, and advances understanding of their properties.

Contribution

It introduces new Diophantine sets extending badly approximable points and deepens the analysis of their relation to Dirichlet improvability and singular points.

Findings

Extended the notion of badly approximable sets.

Clarified the relationship between Dirichlet improvability and Diophantine sets.

Provided new insights into the structure of Diophantine approximation sets.

Abstract

This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the notion of badly approximability.