Tensor trains approximation estimates in the Chebyshev norm

A new elementwise bound on the cross approximation error used for approximating multi-index arrays (tensors) in the format of a tensor train is obtained. The new bound is the first known error bound that differs from the best bound by a factor that depends only on the rank of the approximation $r$ and on the dimensionality of the tensor $d$, and the dependence on the dimensionality at a fixed rank has only the order $d^{\operatorname{const}}$ rather than $\operatorname{const}^d$. Thus, this bound justifies the use of the cross method even for high dimensional tensors.