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Dimension estimates for badly approximable affine forms
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 06 November 2024, pp. 1-56
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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 where
$$ \begin{align*}\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b\rangle^m \geq \epsilon,\end{align*} $$
$\langle \cdot \rangle $ denotes the distance from the nearest integral vector. 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 a Diophantine condition of A equivalent to the full Hausdorff dimension of the set of 
$\epsilon $-badly approximable targets for fixed A. 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.
