We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov operator associated to the rotation component of the isometries has spectral gap. We also prove that certain self-similar measures are absolutely continuous with smooth densities. These families of self-similar measures give higher-dimensional analogues of Bernoulli convolutions on which absolute continuity can be established for contraction ratios in an open set.
Elon Lindenstrauss. Péter P. Varjú. "Random walks in the group of Euclidean isometries and self-similar measures." Duke Math. J. 165 (6) 1061 - 1127, 15 April 2016. https://doi.org/10.1215/00127094-3167490