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Hausdorff dimension of unions of $k$-planes
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abstract
We prove a conjecture of H\'era 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.
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Cited by 1 Pith paper
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On the packing dimension of unions and extensions of $k$-planes
For unions of subsets of k-planes, the paper proves packing dimension bounds analogous to known Hausdorff dimension results, and for hyperplanes proves an exact preservation of packing dimension under full-dimension e...
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