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arXiv · 2305.14544

Hausdorff dimension of unions of $k$-planes

Shengwen Gan
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Abstract

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

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Shengwen Gan. 2023-07-24. Hausdorff dimension of unions of $k$-planes. https://arxiv.org/abs/2305.14544

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