Robust Polynomial Freiman-Ruzsa from Corrupted Set Observations
We study structural recovery from an exact but adversarially corrupted set observation over $\mathbb F_2^n$. A hidden nonempty set $A$ satisfies $|A+A|\leq K|A|$, while the algorithm receives deterministic membership access and independent exact uniform samples only from a set $B$ satisfying $|A\triangle B|\leq\eta|A|$. For $\eta\leq cK^{-1/2}$, we give a randomized FPT-form algorithm which, with high probability, outputs a subspace $V$ satisfying $|V|\leq|A|$ and $\mathcal N_V(A)\leq K^{O(1)}$. For every supplied $\eta<1$, writing $\varepsilon=1-\eta$, we also give an observation-only algorithm that outputs $O(\sqrt K\,\varepsilon^{-2}\log(3/\varepsilon))$ subspaces. For every hidden set compatible with $B,K,\eta$, some list entry has size at most that hidden set and covering number $\operatorname{poly}(K,\varepsilon^{-1})$. The sample complexity is polynomial, while the direct query and running-time bounds are XP. Every nonempty compatibility class also admits, nonconstructively, one common subspace $V$ such that $|V|\leq|A|$ and $\mathcal N_V(A)\leq2K(1-\eta)^{-1}P_{\rm PFR}(K)$ simultaneously for every compatible hidden set $A$. An exact two-subspace construction forces common covering cost $\Theta((1-\eta)^{-1/2})$, leaving quantitative and algorithmic list-to-single gaps. We further show that the $K^{-1/2}$ contamination scale is optimal up to constants for the one-core, size-only lifting mechanism used in the single-output argument. The proofs combine a persistent randomized Balog-Szemer\'edi-Gowers procedure producing a fixed implicit small-doubling subset on the $\sqrt{\alpha}$ retained-mass scale, conditionally exact finite product sampling, size-oblivious algorithmic PFR, and deterministic lifting.