Subspaces of tensors with high analytic rank

Abstract

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

8 pages