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

For (0, \, 1/2] let K_ R be a self-similar set generated by the iterated function system \ x, \, x+1- \. Given x (0, \, 1/2), let (x) be the set of (0, \, 1/2] such that x K_. In this paper we show that (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 (0, \, 1/2] such that y₁, \, , \, yₚ K_.