For a real \(n\times m\) matrix \(\pmb {\xi }\) , we consider its sequence of best Diophantine approximation vectors \( \pmb {x}_i \in \mathbb {Z}^{{m}}, \, i =1,2,3,... \) , the sequences of its norms \(X_i = \Vert \pmb {x}_i\Vert \) and the norms of remainders \(L_i = \Vert \pmb {\xi }\pmb {x}_i\Vert \) . It is known that, in the cases \(m=1\) , bad approximability of \(\pmb {\xi }\) is equivalent to the boundedness of ratios \(\frac{X_{i+1}}{X_i}\) , while for \(n=1\) bad approximability of \(\pmb {\xi }\) is equivalent to the boundedness of ratios \( \frac{L_i}{L_{i+1}}\) . Moreover, carefully constructed example show that in the cases \(m=1\) and \(n=1\) boundedness of ratios \( \frac{L_i}{L_{i+1}}\) and \(\frac{X_{i+1}}{X_i}\) respectively (the order of ratios changed), does not imply bad approximability of \(\pmb {\xi }\) . In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of \(\pmb {\xi }\) , in particular, what restrictions it gives for Diophantine exponents \(\omega (\pmb {\xi })\) and \(\hat{\omega }(\pmb {\xi })\) . One of our particular results deals with the case \(m=n=2\) . We prove that for \(2\times 2 \) matrices \(\pmb {\xi }\) boundedness of both ratios \( \frac{X_{i+1}}{X_i}, \frac{L_i}{L_{i+1}} \) implies inequality \(\hat{\omega }(\pmb {\xi })\leqslant \frac{4}{3}\) and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.
