Grid distortion lower bound for finite \ell_\infty grids in the Schatten trace class

Establish whether there exists a universal constant c>0 such that the least bi-Lipschitz distortion of the grid [⌊√n⌋]_∞^n into the Schatten trace class S_1 satisfies c_{S_1}([⌊√n⌋]_∞^n)≥c√n for every n.

Background

The paper explains that sharp metric cotype scaling yields lower bounds for the distortion of finite grids. For the Schatten trace class S_1, the relevant grid is [⌊√n⌋]∞n equipped with the ℓ∞ metric, and c_{S_1}(A) denotes the infimum of the product of the Lipschitz constants of an injective embedding and its inverse.

The authors identify the lower bound of order √n as a concrete motivation for the sharp-scaling problem. The general metric cotype estimate available before the paper gives only a lower bound of order n{1/6}; although the paper proves sharp metric cotype for S_1, it does not establish the stated grid-distortion bound itself.

References

One concrete motivation is the grid-distortion question in Question~3. Write $[k]\inftyn={0,\ldots,k}n$ with the $\ell\infty$ distance, and let $c_Y(A)$ be the infimum of $|h|{\mathrm{Lip}|h{-1}|{\mathrm{Lip}$ over injective maps $h:A\to Y$. For $Y=S_1$, the question asks whether

c_{S_1}([\lfloor\sqrt n\rfloor]_\inftyn)\geq c\sqrt n

for some $c>0$ independent of $n$.

— The Sharp Scale Inequality for the Schatten Trace Class  (2610.21445 - Jiao et al., 15 Oct 2026) in Section 1, Introduction, paragraph beginning “One concrete motivation is the grid-distortion question”