It was proved by Oberlin and by Falconer and Mattila
that the union of any $s$Hausdorffdimensional family of affine hyperplanes
in $\mathbb R^n$ has Hausdorff dimension $s+n1$ if
$s\in [0,1]$, and positive Lebesguemeasure if $s > 1$.
We generalize the result for the range $s \in [0,1]$, for $k$dimensional
affine subspaces ($1 \leq k \leq n1$)
in place of hyperplanes.
We also prove a lower estimate for the Hausdorff dimension of generalized
Furstenbergtype sets:
sets intersecting every element of a given family of $k$dimensional affine
subspaces in a set of
Hausdorff dimension at least $\alpha$, where $0 < \alpha \leq k$.
We also study the following closely related questions:
What is the smallest possible Hausdorff dimension of a set which contains
the
$k$skeleton of an $n$dimensional axisparallel scaled cube / rotated and
scaled cube /
rotated unit cube around every point of $\mathbb{R}^n$?
Joint work with Alan Chang, Marianna Csörnyei, Tamás Keleti and András
Máthé.
