Hausdorff dimension of unions of affine subspaces and related problems
Ergodic Theory and Dynamical Systems Seminar
19th November 2020, 4:30 pm – 5:30 pm
Online, on zoom (to participate, email the organisers for a link),
We consider the question of how large a union of affine subspaces must be depending on the family of affine subspaces constituting the union. In the famous Kakeya problem one considers lines in every direction. Here the position of the lines or higher-dimensional affine subspaces is more general, and accordingly the expected dimension bound is different. We prove that the union of any s-dimensional family of k-dimensional affine subspaces is at least k + s/(k+1) -dimensional, and is exactly k + s -dimensional if s is at most 1.
Partially based on joint work with Tamás Keleti and András Máthé.