# Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets

### Kornélia Héra

Eötvös Loránd University, Budapest, Hungary### Tamás Keleti

Eötvös Loránd University, Budapest, Hungary### András Máthé

University of Warwick, Coventry, UK

## Abstract

We prove that for any $1≤k<n$ and $s≤1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of $R_{n}$ has Hausdorff dimension $k+s$. More generally, we show that for any $0<α≤k$, if $B⊂R_{n}$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of $R_{n}$ such that every $P∈E$ intersects $B$ in a set of Hausdorff dimension at least $α$, then dim $B≥2α−k+min(dimE,1)$, where dim denotes the Hausdorff dimension. As a consequence, we generalize the well-known Furstenberg-type estimate that every $α$-Furstenberg set has Hausdorff dimension at least $2α$; we strengthen a theorem of Falconer and Mattila [5]; and we show that for any $0≤k<n$, if a set $A⊂R_{n}$ contains the $k$-skeleton of a rotated unit cube around every point of $R_{n}$, or if $A$ contains a $k$-dimensional affine subspace at a fixed positive distance from every point of $R_{n}$, then the Hausdorff dimension of $A$ is at least $k+1$.

## Cite this article

Kornélia Héra, Tamás Keleti, András Máthé, Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets. J. Fractal Geom. 6 (2019), no. 3, pp. 263–284

DOI 10.4171/JFG/77