On simultaneously badly approximable numbers

Let α, β , δ ∈ [0, 1], α + β = 1 and BAD * ( α ,   β ;   δ )   =   { ξ   =   ( ξ 1 ,   ξ 2 )   ∊  [ 0 ,   1 ] 2   :   inf p ∊ N   max {( p log ( p + 1 ) ) α ‖ p ξ 1 ‖ ,   p log( p + 1 ) β ‖ p ξ 2 ‖ } ⩾ δ } . It is proved that for different (α 1 , β 1 ), (α 2 , β 2 ), α 1 + β 1 = α 2 + β 2 = 1 and δ small enough BAD * ( α 1 ,   β 1 ;   δ )   ∩   BAD * ( α 2 ,   β 2 ;   δ )   ≠   Ø . This result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.