Subspaces of tensors with high analytic rank

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 $\Omega_{d,p}(r)$. As an application, we generalize a result of Altman on Szemer\'edi's theorem with random differences.