MiS Preprint Repository

It is known that, in general, the set ${\mathcal{R}}_n$ of tensors of rank $\le n$ is nonclosed. Hence, there are tensors $\mathbf{w}$ in the closure of ${\mathcal{R}}_n$ but not in ${\mathcal{R}}_n$ which are the limits of sequences ${\mathbf{v}}_{\nu }\in {\mathcal{R}}_n.$ Let ${\mathbf{v}}_{\nu }=\sum _{i=1}^n{\mathbf{z}}_{i,\nu }$ be a representation by elementary tensors ${\mathbf{z}}_{i,\nu }.$ It is well-known that ${\delta }_{\nu }:=max\{\Vert {\mathbf{z}}_{i,\nu }\Vert :1\le i\le n\}$ is unbounded as $\nu \to \mathrm{\infty }.$ The error ${\epsilon }_{\nu }:=\Vert \mathbf{w}-{\mathbf{v}}_{\nu }\Vert$ tends to zero. Since ${\delta }_{\nu }\to \mathrm{\infty }$ describes an instability, it is of interest how ${\delta }_{\nu }$ depends on ${\epsilon }_{\nu }.$ In a previous paper the author proved for $n=2$ that both quantities are related by ${\delta }_{\nu }\ge O({\epsilon }_{\nu }^{-1/2}).$ This article concerns the case of $n=3$ and proves ${\delta }_{\nu }\ge O({\epsilon }_{\nu }^{-1/3}).$