For a norm $F$ on $R^2$, we consider the set of $F$-Dirichlet improvable numbers. In case of $F$ a supremum norm, it is well-known that the corresponding set of Drichlet improvable numbers is equal to the union of badly approximable numbers with rational numbers. It is also known that both badly approximable numbers and each $F$-Dirichlet improvable numbers are of measure zero and of full Hausdorff dimension.
Using the classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of Dirichlet improvable numbers with respect to $L_p$ norm in terms of the occurrence of patterns in regular continued fraction expansions.
As an application, we answer two open questions by Kleinbock and Rao by showing that the set of $p$-Dirichlet improvable numbers, which are not badly approximable, is of full Hausdorff dimension. Similarly, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ($p=2$) minus Dirichlet improvable numbers in taxicab norm ($p=1$) and vice versa, are of full Hausdorff dimension.
This is a joint work with Nikolay Moshchevitin.
