Even unique intersection point can break OSC: an example

Abstract

This was a long-standing question since 90-ies whether one-point intersection property for a self-similar set implies open set condition. We answer this question negatively. We give an example of a totally disconnected self-similar set $K\subset \mathbb R$ which does not have open set condition and has minimal overlap of its pieces, that is, all intersections of its pieces $K_i\cap K_j,i\neq j$ are empty except only one, which is a single point.