Kornélia Héra: Furstenberg-type estimates for unions of affine subspaces

A compact plane set is a t-Furstenberg set for some t in (0,1), if it has an at least t-dimensional intersection with some line in each direction (here and in the sequel dimension refers to Hausdorff dimension). Classical results are that every t-Furstenberg set has dimension at least 2t, and at least t + 1/2.
We generalize these estimates for families of affine subspaces. As a result, 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.
Joint work with Tamás Keleti and András Máthé.