🤖 AI Summary
This work addresses the problem of recovering hidden additive structure from a set perturbed by adversarial noise, where the symmetric difference between the observed and true sets is bounded yet potentially destructive to the original structure. We introduce the first composable Balog–Szemerédi–Gowers (BSG) compiler interface that requires no prior commitments, supports persistent membership queries, admits exact finite sampling, and preserves sharp quality guarantees. Our approach integrates an algorithmic polynomial Freiman–Ruzsa theorem without size priors, deterministic amplification, and adaptive product sampling. When η = O(K⁻¹/²), the algorithm outputs a subspace V in FPT time such that |V| ≤ |A| and 𝒩_V(A) ≤ K^O(1). For any η < 1, it produces—in polynomial sample complexity and XP time—a common list covering all hidden sets compatible with the observations.
📝 Abstract
We study structural inference from an exact, 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 and exact uniform-sampling access only to $B$, where $|A\triangle B|\leqη|A|$. Since corruption can destroy the doubling of $B$, sharp existential BSG and clean-input Algorithmic PFR do not directly compose in this model.
Our main result is a promise-free sharp persistent-subset BSG compiler. Given sample-and-query access to $S$ and $α$, it returns either $\mathsf{FAIL}$ or a descriptor defining one fixed subset $Y\subseteq S$ with $|Y|\geq c\sqrtα\,|S|$ and $|Y+Y|\leq Cα^{-4}|Y|$. Without an energy promise, every nonfailure output is valid except with the prescribed soundness probability; high energy guarantees success with high probability. The descriptor gives persistent membership under adaptive queries, and a finite-horizon bridge gives conditionally exact product samples. Thus sharp retained mass, promise-free validity, persistence, and exact finite sampling form one composable interface.
Combined with certified size-oblivious Algorithmic PFR and deterministic lifting, the compiler yields a randomized FPT-form algorithm for $η=O(K^{-1/2})$, outputting $V$ with $|V|\leq|A|$ and $\mathcal N_V(A)\leq K^{O(1)}$. For every supplied $η<1$, an iterated residual algorithm outputs a common list serving every compatible hidden set with polynomial covering budget; samples are polynomial, while membership-query and running-time complexity are XP. Finally, every nonempty compatibility class admits one common subspace nonconstructively, whereas an exact two-subspace construction forces common covering cost $Θ((1-η)^{-1/2})$.