Kornélia Héra

Title: Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces

Abstract: A plane set is called 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 $t + 1/2$.

Given integers $0 < k < n$, a family E of $k$-dimensional affine subspaces in $\mathbb R^n$, and $0 < t < k$, we say that a set $A$ in $\mathbb R^n$ is a $(t,E)$-Furstenberg set, if $A$ has an at least $t$-dimensional intersection with each $k$-dimensional affine subspace of the family $E$. We give dimension estimates for $(t,E)$-Furstenberg sets, which are generalizations of the classical plane results. As a consequence, 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.

The talk is partially based on joint work with Tamás Keleti and András Máthé.