Badly approximable vectors on a vertical Cantor set

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^qN,^maxqqx1/i, q^qy¹/j^>c^, where ||^x|| is the distance from 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 (i_t, j_t) with i_t, j_t > 0, i_t + j_t = 1. Then, the intersection of Bad(i_t, j_t) over all t is nonempty.