Multiplicatively badly approximable matrices up to logarithmic factors

Abstract

Let $\Vert x\Vert$ denote the distance from $x\in \mathbb{R}$ to the nearest integer. In this paper, we prove an existence and density statement for matrices $\mathbf{A}\in {\mathbb{R}}^{m\times n}$ satisfying $$\underset{|\mathbf{q}{|}_{\infty }\to +\infty }{\mathrm{liminf}}\prod _{j=1}^n\max \{1,|q_j|\}\log {\left(\prod _{j=1}^n\max \{1,|q_j|\}\right)}^{m+n-1}\prod _{i=1}^m\Vert A_i\mathbf{q}\Vert >0,$$ where the vector $\mathbf{q}$ ranges in ${\mathbb{Z}}^n$ and $A_i$ are the rows of the matrix $\mathbf{A}$. 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.

References in corpus (1)

• On multiplicatively badly approximable numbers