2026/07/16 by Balázs Bárány, Antti Käenmäki, István Kolossváry
#math.CA #math.DS
We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in ℝd. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set X itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if X additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.