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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 & 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.
Original language  English 

Article number  108316 
Number of pages  57 
Journal  Advances in Mathematics 
Volume  401 
Early online date  17 Mar 2022 
DOIs  
Publication status  Published  4 Jun 2022 
Bibliographical note
© 2022 The Authors.Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research