Subspaces of tensors with high analytic rank

Jop Briët

备注：: 8 pages

英文摘要：

It is shown that for any subspace $V\subseteq \mathbb{F}_p^{n\times\cdots\times n}$ of $d$-tensors, if $\dim(V) \geq tn^{d-1}$, then there is subspace $W\subseteq V$ of dimension at least $t/(dr) - 1$ whose nonzero elements all have analytic rank $Ω_{d,p}(r)$. As an application, we generalize a result of Altman on Szemerédi's theorem with random differences.

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