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Multiplicatively badly approximable matrices up to logarithmic factors

Part of: Diophantine approximation, transcendental number theory Geometry of numbers

Published online by Cambridge University Press:  17 May 2021

REYNOLD FREGOLI
REYNOLD FREGOLI*
Affiliation:
Department of Mathematics, Bedford Building, Royal Holloway University of London, Egham Hill, TW20 0EX UK e-mail: Reynold.Fregoli.2017@live.rhul.ac.uk
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Abstract

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.

MSC classification

Primary: 11J13: Simultaneous homogeneous approximation, linear forms 11J20: Inhomogeneous linear forms 11J83: Metric theory 11H46: Products of linear forms
Secondary: 11H16: Nonconvex bodies
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 685 - 703
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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