Dirichlet improvability in $L_p$-norms

Abstract

For a norm $F$ on ${\mathbb{R}}^2$, we consider the set of $F$-Dirichlet improvable numbers ${\mathbf{DI}}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\infty$, which is a supremum norm, it is well-known that ${\mathbf{DI}}_F=\mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\mathbf{BA}$ and each ${\mathbf{DI}}_F$ are of measure zero and of full Hausdorff dimension. Using classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of ${\mathbf{DI}}_p:={\mathbf{DI}}_{F^{[p]}}$ in terms of the occurrence of patterns in regular continued fraction expansions, where $F^{[p]}$ is an $L_p$-norm with $p\in [1,\infty )$. This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set ${\mathbf{DI}}_p\setminus \mathbf{BA}$ is of full Hausdorff dimension, as well as proving some results about the size of the difference ${\mathbf{DI}}_{p_1}\setminus {\mathbf{DI}}_{p_2}$. To be precise, 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, that is ${\mathbf{DI}}_2\setminus {\mathbf{DI}}_1$ and ${\mathbf{DI}}_1\setminus {\mathbf{DI}}_2$, are of full Hausdorff dimension. We also find all values of $p$, for which the set ${\mathbf{DI}}_p^c\cap \mathbf{BA}$ has full Hausdorff dimension. Finally, our characterization result implies that the number $e$ satisfies $e\in {\mathbf{DI}}_p$ if and only if $p\in (1,2)\cup (p_0,\infty )$ for some special constant $p_0\approx 2.57$.

34 pages plus 13 pages long Appendix, any comments are appreciated