Bad approximability, bounded ratios and Diophantine exponents

Abstract For a real $$n\times m$$ n × m matrix $$\pmb {\xi }$$ ξ , we consider its sequence of best Diophantine approximation vectors $$ \pmb {x}_i \in \mathbb {Z}^{{m}}, \, i =1,2,3,... $$ x i ∈ Z m , i = 1 , 2 , 3 , . . . , the sequences of its norms $$X_i = \Vert \pmb {x}_i\Vert $$ X i = ‖ x i ‖ and the norms of remainders $$L_i = \Vert \pmb {\xi }\pmb {x}_i\Vert $$ L i = ‖ ξ x i ‖ . It is known that, in the cases $$m=1$$ m = 1 , bad approximability of $$\pmb {\xi }$$ ξ is equivalent to the boundedness of ratios $$\frac{X_{i+1}}{X_i}$$ X i + 1 X i , while for $$n=1$$ n = 1 bad approximability of $$\pmb {\xi }$$ ξ is equivalent to the boundedness of ratios $$ \frac{L_i}{L_{i+1}}$$ L i L i + 1 . Moreover, carefully constructed example show that in the cases $$m=1$$