Title: Slices, projections and tangents of planar self-affine sets
Abstract: Weak tangents are sets arising as limits under successive magnifications into a compact set. For dominated self-affine sets, the distortion caused by the affine maps results in a fiber structure for the weak tangents, and under various separation assumptions on the underlying iterated function system, it has been shown that the Assouad dimension—the Hausdorff dimension of the largest weak tangent—is characterized by the dimensions of certain slices and projections of the self-affine set.
In this talk, I will present a slicing theorem for weak tangents of self-affine sets which requires no separation assumptions and only assumes weak domination of the matrix parts. Under a separation condition which is slightly weaker than the strong separation condition, we are further able to connect the slices of weak tangents with the slices of the set itself. My aim is to convey the main ideas behind the proof, which is at its essence a discretized pigeonhole argument. The proof is entirely self-contained and, unlike previous work on the topic, we do not use the deep machinery of Bárány-Hochman-Rapaport or need any assumptions on the regularity of the projections of the set. The talk is based on joint work with Alex Rutar.
