Balázs Bárány

Projection of self-affine sets onto lines

In this talk, we show an all-directions Marstrand–Mattila projection theorem for self-affine measures and sets in R^d. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line is exact dimensional and 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. As a corollary, if a self-affine set 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. This is a joint work with Antti Käenmäki and István Kolossváry.