On the closure of a class of subsets of the real line

A subset, X , of R belongs to the family T ( a ) if | X | ≥ 2 and, for all x and y belonging to X, ax + (1 − a ) y ∈ X , where a ∈ R . I consider the problem of determining for which values of a > 1 all elements of T ( a ) are dense in R .