For i,j>0,i+j=1, the set of badly approximable vectors with weight
(i,j) is defined by Bad(i,j)={(x,y)∈R2:∃c>0∀q∈N,max{q∣∣qx∣∣1/i,q∣∣qy∣∣1/j}>c}, where ∣∣x∣∣ is the
distance of x to the nearest integer. In 2010 Badziahin-Pollington-Velani
solved Schmidt's conjecture which was stated in 1982, proving that Bad(i,j)∩Bad(j,i) is nonempty. Using Badziahin-Pollington-Velani's technique with
reference to fractal sets, we were able to improve their results: Assume that
we are given a sequence (it,jt) with it,jt>0,it+jt=1. Then,
the intersection of Bad(it,jt) over all t is nonempty