When is the union of two unit intervals a self-similar set satisfying the open set condition?

Let U λ be the union of two unit intervals with gap λ. We show that U λ is a self-similar set satisfying the open set condition if and only if U λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations of the set of all such λ’s, as well as the corresponding tiling patterns, are given.