The interplay between bounded ranks of tensors arising from partitions

Let be integers. Using a fragmentation technique, we characterise ‐tuples of non‐empty families of partitions of such that it suffices that an order‐ tensor has bounded ‐rank for each for it to have bounded ‐rank. On the way, we prove power lower bounds on suitable products of diagonal tensors, providing a qualitative answer to a question of Naslund.