论文标题
近似向量的几何和算术方面
Geometric and arithmetic aspects of approximation vectors
论文作者
论文摘要
令$θ\ in \ mathbb {r}^d $。 We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $θ$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the approximating vector $(p,q)$;位移矢量$（p -qθ）$;以及$（d + 1）$ - 元组$（p，q）$ modulo的零件的残留类别。所有这些都已与二芬太汀近似问题有关。我们将所有这些数量的渐近分布视为$（p，q）$范围比最佳近似物和$ε$ - $θ$的含量范围，并描述了相关空间的限制措施，这是Lebesgue A.E.的限制措施。 $θ$。我们还考虑了对向量的$θ$的类似问题，其组件及1个完全实真实的$ d+1 $的实际数字字段。我们的技术涉及将问题重新塑造为在Adelic空间上单参数流的横截面的等分分配问题，这是在$（D + 1）$ - 尺寸晶格的空间上的振动。我们的结果将许多以前的作者的结果推广到更高的维度和均等分配。
Let $θ\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $θ$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the approximating vector $(p,q)$; the displacement vector $(p - qθ)$; and the residue classes of the components of the $(d + 1)$-tuple $(p, q)$ modulo all primes. All of these have been studied in connection with Diophantine approximation problems. We consider the asymptotic distribution of all of these quantities, properly rescaled, as $(p, q)$ ranges over the best approximants and $ε$-approximants of $θ$, and describe limiting measures on the relevant spaces, which hold for Lebesgue a.e. $θ$. We also consider a similar problem for vectors $θ$ whose components, together with 1, span a totally real number field of degree $d+1$. Our technique involve recasting the problem as an equidistribution problem for a cross-section of a one-parameter flow on an adelic space, which is a fibration over the space of $(d + 1)$-dimensional lattices. Our results generalize results of many previous authors, to higher dimensions and to joint equidistribution.
