Speaker: Antoine Marnat
4 November, 2020 14:30 UTC
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 disjoint union is not the full set of Dirichlet improvable vectors: we prove that there exist uncountably many Dirichlet improvable vectors that are neither badly approximable nor singular. We construct these numbers using the parametric geometry of numbers. Furthermore, by doing so we can choose the exponent of Diophantine approximation by a rational subspace of dimension exactly $d$, for any d between $0$ and $n-1$.