论文标题
乘以近似矩阵，直到对数因素
Multiplicatively badly approximable matrices up to logarithmic factors
论文作者
论文摘要
令$ \ | x \ | $表示距离$ x \ in \ mathbb {r} $到最近整数的距离。在本文中，我们证明了矩阵的存在和密度语句$ \ boldsymbol {a} \ in \ mathbb {r}^{m \ times n} $满足$ \ liminf_ {| \ boldsymbol +\ \ infty} \ prod_ {j = 1}^{n} \ max \ {1，| q_ {j} | \} | \ log \ left（\ prod_ = 1} {1，| q_ {j} | \} \ right）^{m+n-1} \ prod_ {i = 1}^{m}^{m} \ | a_ {i} \ boldsymbol {q} \ |> 0，$$ vector $ \ boldsymbol {q} $范围$ \ mathbb {z}^{n} $和$ a_ {i} $是matrix $ \ boldsymbol {a a} $的行。该结果将Moshchevitin的先前结果以$ 2 $维的向量扩展到任意维度。当前尚不可用，将Moshchevitin方法应用于$ M> 2 $所需的估计值。因此，我们开发了一种基本不同的方法，使我们能够克服这个问题。我们还将这种存在的结果推广到不均匀的环境。具有上述特性的矩阵似乎具有很小的分数零件的倒数。这一事实有助于我们阐明Lêand Vaaler提出的一个问题，从而证明了在较高维度中对此类总和的一些新估计。
Let $\|x\|$ denote the distance from $x\in\mathbb{R}$ to the nearest integer. In this paper, we prove an existence and density statement for matrices $\boldsymbol{A}\in\mathbb{R}^{m\times n}$ satisfying $$\liminf_{|\boldsymbol{q}|_{\infty}\to +\infty}\prod_{j=1}^{n}\max\{1,|q_{j}|\}\log\left(\prod_{j=1}^{n}\max\{1,|q_{j}|\}\right)^{m+n-1}\prod_{i=1}^{m}\|A_{i}\boldsymbol{q}\|>0,$$ where the vector $\boldsymbol{q}$ ranges in $\mathbb{Z}^{n}$ and $A_{i}$ are the rows of the matrix $\boldsymbol{A}$. This result extends a previous result of Moshchevitin for $2$-dimensional vectors to arbitrary dimension. The estimates needed to apply Moshchevitin's method to the case $m>2$ are not currently available. We therefore develop a substantially different method, that allows us to overcome this issue. We also generalise this existence result to the inhomogeneous setting. Matrices with the above property appear to have a very small sum of reciprocals of fractional parts. This fact helps us to shed light on a question raised by Lê and Vaaler, thereby proving some new estimates for such sums in higher dimension.
