  For $\alpha, \beta, \delta \in [0,1], \alpha +\beta = 1 $ we consider sets $$
{\rm BAD}^* (\alpha, \beta ;\delta) = \left\{\xi = (\xi_1,\xi_2) \in [0,1]^2:
,\inf_{p\in \mathbb{N}} \max \{(p\log(p+1))^\alpha ||p\xi_1||,
  (p\log (p+1))^\beta ||p\xi_2||\} \ge \delta \right\}. $$ We prove that for
different $(\alpha_1,\beta_1), (\alpha_2,\beta_2), \alpha_1 +\beta_1 = \alpha_2
+\beta_2 = 1 $ and $\delta $ small enough $$ {\rm BAD}^* (\alpha_1, \beta_1
;\delta) \bigcap {\rm BAD}^* (\alpha_2, \beta_2 ;\delta) \neq \varnothing . $$
  Our result is based on A. Khintchine's construction and an original method
due to Y. Peres and W. Schlag.
