Bad approximability, bounded ratios and Diophantine exponents

For a real n× m n × m matrix ξ ξ , we consider its sequence of best Diophantine approximation vectors xᵢ ∈ Z^m, \, i =1,2,3,... x i ∈ Z m , i = 1 , 2 , 3 , . . . , the sequences of its norms Xᵢ = xᵢ X i = ‖ x i ‖ and the norms of remainders Lᵢ = ξ xᵢ L i = ‖ ξ x i ‖ . It is known that, in the cases $$m=1$$ m = 1 , bad approximability of ξ ξ is equivalent to the boundedness of ratios Xᵢ₊₁Xᵢ X i + 1 X i , while for $$n=1$$ n = 1 bad approximability of ξ ξ is equivalent to the boundedness of ratios LᵢLᵢ₊₁ L i L i + 1 . Moreover, carefully constructed example show that in the cases $$m=1$$</jats:tex-m