Inhomogeneous Diophantine approximation for generic homogeneous functions

Abstract

The present paper is a sequel to [Monatsh.~Math.\ {\bf 194} (2021), 523--554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers $n \geq 2$ and $\ell \geq 1$, any ${\pmb ξ} = \left(ξ_1, \dots , ξ_\ell \right) \in \mathbb{R}^\ell$, and any homogeneous function \linebreak $f = \left(f_1, \dots , f_\ell \right): \mathbb{R}^n \to \mathbb{R}^\ell$ that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function $ψ= \left(ψ_1, \dots , ψ_\ell \right): \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ for a generic element $f \circ g$ in the $\operatorname{SL}_n(\mathbb{R})$-orbit of $f$ to be (respectively, not to be) $ψ$-approximable at ${\pmb ξ} = (ξ_1,\dots,ξ_n)$: that is, for there to exist infinitely many (respectively, only finitely many) $\mathbf{v} \in \mathbb{Z}^n$ such that $\left|ξ_j - \left( f_j \circ g\right)(\mathbf{v})\right| \leq ψ_j(\|\mathbf{v}\|)$ for each $j \in \left\lbrace 1, \dots, \ell \right\rbrace$. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of $f$ that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace $\operatorname{SL}_n(\mathbb{R})$ above by any closed subgroup of $\operatorname{ASL}_n(\mathbb{R})$ that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper.