Let \[||x||\] denote the distance from \[x \in \mathbb{R}\] to the nearest integer. In this paper, we prove a new existence and density result for matrices \[A \in {\mathbb{R}^{m \times n}}\] satisfying the inequality

\[\mathop {\lim \inf }\limits_{|q{|_\infty } \to + \infty } \prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} \log {\left( {\prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} } \right)^{m + n - 1}}\prod\limits_{i = 1}^m {{A_i}q} > 0,\]

where q ranges in \[{\mathbb{Z}^n}\] and Ai denote the rows of the matrix A. This result extends previous work of Moshchevitin both to arbitrary dimension and to the inhomogeneous setting. The estimates needed to apply Moshchevitin’s method to the case m > 2 are not currently available. We therefore develop a substantially different method, based on Cantor-like set constructions of Badziahin and Velani. Matrices with the above property also appear to have very small sums of reciprocals of fractional parts. This fact helps us to shed light on a question raised by Lê and Vaaler on such sums, thereby proving some new estimates in higher dimension.

令\[||x||\]表示从\[x \in \mathbb{R}\]到最接近的整数的距离。在本文中，我们证明了满足不等式的矩阵\[A \in {\mathbb{R}^{m \times n}}\]的新存在性和密度结果

\[\mathop {\lim \inf }\limits_{|q{|_\infty } \to + \infty } \prod\limits_{j = 1}^n {\max } \{ 1,|{q_j} |\} \log {\left( {\prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} } \right)^{m + n - 1}}\ prod\limits_{i = 1}^m {{A_i}q} > 0,\]

其中q在\[{\mathbb{Z}^n}\]中的范围和A i表示矩阵A的行。这一结果将 Moshchevitin 先前的工作扩展到任意维度和非均匀设置。将 Moshchevitin 方法应用于m > 2 的情况所需的估计值目前不可用。因此，我们基于 Badziahin 和 Velani 的 Cantor 式集合构造开发了一种截然不同的方法。具有上述性质的矩阵似乎也具有非常小的小数部分倒数之和。这一事实有助于我们阐明 Lê 和 Vaaler 就此类总和提出的问题，从而在更高维度上证明了一些新的估计。