2025/05/21 by Antoine Marnat, Nikolay Moshchevitin, Marnat, Antoine +4
#math.NT
paper · pdf · doi:10.48550/arxiv.2505.15964
For a real m× n matrix \pmbξ, we consider its sequence of best Diophantine approximation vectors \pmbxi ∈ ℤn, i =1,2,3, ... , the sequences of its norms Xi = ‖\pmbxi‖ and the norms of remainders Li = ‖\pmbξ\pmbxi‖. It is known that, in the cases m=1, bad approximability of \pmbξ is equivalent to the boundedness of ratios \fracXi+1Xi, while for n=1 bad approximability of \pmbξ is equivalent to the boundedness of ratios \fracLiLi+1. Moreover, carefully constructed example show that in the cases m=1 and n=1 boundedness of ratios \fracLiLi+1 and \fracXi+1Xi respectively (the order of ratios changed), does not imply bad approximability of \pmbξ. In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of \pmbξ, in particular, what restrictions it gives for Diophantine exponents ω(\pmbξ) and ω(\pmbξ). One of our particular results deals with the case m=n=2. We prove that for 2× 2 matrices \pmbξ boundedness of both ratios \fracXi+1Xi, \fracLiLi+1 implies inequality ω(\pmbξ)≤ (4)/(3) and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.