Inhomogeneous Diophantine approximation for generic homogeneous functions

TL;DR

This paper extends inhomogeneous Diophantine approximation results to generic homogeneous functions, providing criteria for approximation and uniform approximation, and explores examples beyond nonsingularity assumptions.

Contribution

It generalizes previous homogeneous results to inhomogeneous settings and introduces criteria for generic orbits under SL(n,R) actions to be approximable.

Findings

Established biconditional criteria for inhomogeneous approximation

Provided sufficient conditions for uniform approximation

Extended results to functions not satisfying nonsingularity

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 ${\pmb \xi} = \left(\xi_1, \dots , \xi_\ell \right) \in \mathbb{R}^\ell$, 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 $\psi = \left(\psi_1, \dots , \psi_\ell \right): \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ for a generic element $f \circ g$ in the $\operatorname{SL}_n(\mathbb{R})$-orbit of $f$ to be (respectively, not to be) $\psi$-approximable at ${\pmb \xi} = (\xi_1,\dots,\xi_n)$: that is, for there to exist infinitely many (respectively, only finitely many) $\mathbf{v} \in \mathbb{Z}^n$ such that $\left|\xi_j - \left( f_j \circ g\right)(\mathbf{v})\right| \leq \psi_j(\|\mathbf{v}\|)$ 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.