Subspaces of tensors with high analytic rank

Abstract

It is shown that for any subspace V\xe2\x8a\x86Fn\xc3\x97\xe2\x8b\xaf\xc3\x97np of d-tensors, if dim(V)\xe2\x89\xa5tnd\xe2\x88\x921, then there is subspace W\xe2\x8a\x86V of dimension at least t/(dr)\xe2\x88\x921 whose nonzero elements all have analytic rank \xce\xa9d,p(r). As an application, we generalize a result of Altman on Szemer\xc3\xa9di\'s theorem with random differences