Subspaces of tensors with high analytic rank

It is shown that if V ⊆ $F_p^n$ ×⋯×np is a subspace of d-tensors with dimension at least tn^d-1, then there is a subspace W ⊆ V of dimension at least t/(dr)−1 p is a subspace of d-tensors with dimension 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.