Focused Width in Adversarial Fake Detection: A Separation

We study the adversarial fake detection model introduced by Mendelson, Paouris and Vershynin. In this model, a genuine sample is $\pmb{X}\sim N(0,\pmb{I}_n)$, while a fake sample is produced as $\pmb{X}+r\pmb{t}({\pmb{X}})$, where the adversary first observes $\pmb{X}$ and then chooses an admissible perturbation $\pmb{t}({\pmb{X}})$ from a prescribed set $\mathscr{T}\subset\mathbb{R}^n$. The central quantity is the detectability radius $r(\mathscr{T})$, which formalizes the transition scale at which fake samples become reliably distinguishable from genuine ones. Mendelson, Paouris and Vershynin introduced the focused width $\widetilde{w}(\mathscr{T})$ as a geometric parameter for this radius and conjectured that, for every origin-symmetric set $\mathscr{T}$, it characterizes $r(\mathscr{T})$ up to universal constants.
In this note, we disprove this conjecture for a broad class of discrete sets. More precisely, we consider any origin-symmetric set $\mathscr{T}_n$ lying between the hypercube and the odd integer grid: \begin{equation*} \{-1,1\}^n\subset\mathscr{T}_n\subset ( 2\mathbb{Z}+1)^n. \end{equation*} For every such $\mathscr{T}_n$, we prove that $\frac{\widetilde{w}(\mathscr{T}_n)}{r(\mathscr{T}_n) }\gtrsim \sqrt{\log n}$. Thus, in the Gaussian model, the focused width can overestimate the detectability radius by a $\sqrt{\log n}$ factor and therefore does not characterize it in general. We further show that this logarithmic scale is not intrinsic: in the corresponding non-Gaussian model with product Laplace data, the focused width benchmark can even exceed the detectability radius by at least a polynomial factor of order $n^{1/4}$.