On the union of homogeneous symmetric Cantor set with its translations

Fix a positive integer N and a real number 0< β < 1/(N+1) . Let Γ be the homogeneous symmetric Cantor set generated by the IFS {ϕ _i(x)=β x + i 1-β/N: i=0,1,… , N }. For m∈ℤ_+ we show that there exist infinitely many translation vectors t=(t_0,t_1,… , t_m) with 0=t_0