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It is shown that if V \xe2\x8a\x86
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\n\xc3\x97\xe2\x8b\xaf\xc3\x97np is a subspace of d-tensors with dimension at least tnd-1, then there is a subspace W \xe2\x8a\x86 V of dimension at least t/(dr)\xe2\x88\x921
\np is a subspace of d-tensors with dimension 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
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