Dirichlet is not just Bad and Singular

TL;DR

This paper demonstrates that in higher dimensions, the set of Dirichlet improvable vectors is more complex than previously thought, including many that are neither badly approximable nor singular, extending classical Diophantine approximation results.

Contribution

It proves the existence of continuum many Dirichlet improvable vectors that are neither badly approximable nor singular in higher dimensions, and introduces intermediate Dirichlet improvable sets.

Findings

Existence of many Dirichlet improvable vectors outside classical categories.

Extension of Davenport-Schmidt theorem to intermediate dimensions.

Equivalence of intermediate Diophantine sets for various approximation types.

Abstract

It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport and Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main "continuum" result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dircihlet improvable points.