Dirichlet improvability in $L_p$-norms

TL;DR

This paper characterizes Dirichlet improvable numbers in $L_p$-norms using continued fractions, resolving open questions about their size and differences across norms, and identifies conditions for specific constants like e.

Contribution

It provides a complete, effective classification of $L_p$-norm Dirichlet improvable numbers, answering open questions and analyzing their Hausdorff dimensions.

Findings

Set differences $ extbf{DI}_2 ackslash extbf{DI}_1$ and vice versa are of full Hausdorff dimension.

The set $ extbf{DI}_p ackslash extbf{BA}$ has full Hausdorff dimension.

Number $e$ is in $ extbf{DI}_p$ iff $p otin (2, p_0)$ for a constant $p_0 \

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 $p_0\approx2.57$.