On a class of self-similar sets which contain finitely many common points

For λ ∈ (0,\,1/2] let K_λ ⊂ R be a self-similar set generated by the iterated function system \λ x,\, λ x+1-λ \ . Given x∈ (0,\,1/2) , let Λ (x) be the set of λ ∈ (0,\,1/2] such that x∈ K_λ . In this paper we show that Λ (x) is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any y₁,\,…,\, yₚ∈ (0,\,1/2) there exists a full Hausdorff dimensional set of λ ∈ (0,\,1/2] such that y₁,\,…,\, yₚ ∈ K_λ .