Hausdorff dimension of unions of $k$-planes

Abstract

We prove a conjecture of Héra on the dimension of unions of $k$-planes. Let $0<k\le d<n$ be integers, and $\beta \in [0,k+1)$. If $\mathcal{V}\subset A(k,n)$, with $\text{dim}(\mathcal{V})=(k+1)(d-k)+\beta$, then $\text{dim}({\bigcup }_{V\in \mathcal{V}}V)\ge d+\min \{1,\beta \}$. The proof combines a recent idea of Zahl and the Brascamp-Lieb inequality.

arXiv comment15 pages; the range of p in Theorem 5, Theorem 7 and Theorem 8 is corrected