Subspaces of tensors with high analytic rank

Abstract

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