Dimension estimates for badly approximable affine forms

Abstract

For given $\epsilon >0$ and $b\in {\mathbb{R}}^m$, we say that a real $m\times n$ matrix $A$ is $\epsilon$-badly approximable for the target $b$ if $$\underset{q\in {\mathbb{Z}}^n,\Vert q\Vert \to \infty }{\mathrm{liminf}}\Vert q{\Vert }^n⟨Aq-b{⟩}^m\ge \epsilon ,$$ where $⟨\cdot ⟩$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $\epsilon$-badly approximable matrices for fixed target $b$ and the set of $\epsilon$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $\epsilon$-badly approximable targets for fixed $A$ has full Hausdorff dimension for some $\epsilon >0$. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the $A$-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.

50 pages, 1 figure. This paper supersedes the posting arXiv:1904.07476, making the latter obsolete. v1->v2: Appendix is removed, Section 2 is revised