On dense intermingling of exact overlaps and the open set condition

Abstract We prove that certain families of homogenous affine iterated function systems in $\mathbb {R}^{d}$ have the property open set condition and existence exact overlaps both occur densely space translation parameters. These examples demonstrate theorems Falconer Jordan–Pollicott–Simon on almost sure dimensions self-affine sets measures, exceptional parameters can be a dense set. The proof combines results from literature tilings with an adaptation classic argument Erdős singularity Bernoulli convolutions. This result encompasses one-dimensional example due to Kenyon which arises as special case.