论文标题
通用均匀函数的不均匀双苯胺近似
Inhomogeneous Diophantine approximation for generic homogeneous functions
论文作者
论文摘要
本文是[Monatsh。〜Math。\ {\ bf 194}（2021），523--554]的续集，其中该论文的结果被概括，因此它们在不均匀的双养生近似值的情况下保持。给定任何整数$ n \ geq 2 $和$ \ ell \ geq 1 $，任何$ {\pmbξ} = \ left（ξ_1，\ dots，dots，ξ_\ ell \ right \ right \ right） \ Mathbb {r}^n \ to \ Mathbb {r}^\ ell $满足一定的非语言假设，我们在近似函数$ψ= \ left（ψ_1，\ dots，dots，d dots，ψ_\ ell \ right \ right）上获得了双条件标准： \ left（\ Mathbb {r} _ {> 0} \ right）^\ ell $，对于$ \ propatatorName {sl} _n（\ Mathbb {r}）$ f $的通用元素$ f \ circ g $ in $ \ operatatOrname {sl} _n（\ mathbb {r}）$ f $（分别为$ - $ - $ - $ - $ - （ξ_1，\ dots，ξ_n）$：也就是说，在那里，无限存在（分别只有很多）$ \ mathbf {v} \ in \ mathbb {z}^n $，以便$ \ weft | weft | weft | eCCemjj- \ weft（f_j \ circ g \ circ g \ right）（\ right） \ leqψ_j（\ | \ | \ Mathbf {v} \ |）$ in \ left \ left \ lbrace 1，\ dots，\ ell \ right \ rbrace $。在这种情况下，我们还获得了足够的均匀近似条件。我们还考虑了一些$ f $的示例，这些示例无法满足我们的非语言假设，并证明了这些示例的类似结果。 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.
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.
