Subspaces of tensors with high analytic rank

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 \( tn^{d-1} \), then there is a 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.