Dirichlet is not just bad and singular

TY - JOUR

T1 - Dirichlet is not just bad and singular

AU - Beresnevich, Victor

AU - Guan, Lifan

AU - Marnat, Antoine

AU - Ramirez, Felipe Alberto

AU - Velani, Sanju

N1 - © 2022 The Authors.

PY - 2022/6/4

Y1 - 2022/6/4

N2 - 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 & 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 Dirichlet improvable points.

AB - 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 & 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 Dirichlet improvable points.

U2 - 10.1016/j.aim.2022.108316

DO - 10.1016/j.aim.2022.108316

M3 - Article

SN - 0001-8708

VL - 401

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108316

ER -