Separation properties for self-similar sets

Abstract

Given a self-similar set K in R s {\mathbb {R}^s} we prove that the strong open set condition and the open set condition are both equivalent to H α ( K ) > 0 {H^\alpha }(K) > 0 , where α \alpha is the similarity dimension of K and H α {H^\alpha } denotes the Hausdorff measure of this dimension. As an application we show for the case α = s \alpha = s that K possesses inner points iff it is not a Lebesgue null set.