Hausdorff measure of sets of Dirichlet non-improvable affine forms

Abstract

For a decreasing real valued function $\psi$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in {\mathbb{R}}^m$ is said to be $\psi$-Dirichlet improvable if the system $$\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}{\Vert }^m<\psi (T)\text{and}\Vert \mathbf{q}{\Vert }^n<T$$ has a solution $\mathbf{p}\in {\mathbb{Z}}^m$, $\mathbf{q}\in {\mathbb{Z}}^n$ for all sufficiently large $T$, where $\Vert \cdot \Vert$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $\psi$-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the $\psi$-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that $\mathbf{b}$ is fixed. As an application, we compute the Hausdorff dimension of the set of pairs $(A,\mathbf{b})$ with uniform Diophantine exponents $\hat{w}(A,\mathbf{b})\le w$.

arXiv comment35 pages, proof of Lemma 4.6 is revised