  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 L\^e and Vaaler, thereby proving some new
estimates for such sums in higher dimension.
