Polynomial Freiman–Ruzsa Conjecture (F2^n setting)

Establish that if A ⊆ F2^n is nonempty and satisfies |A + A| ≤ C|A|, then A is covered by poly(C) affine subspaces, each of size at most |A|.

Background

In additive combinatorics over F2n, the conjecture asserts strong structural control of sets with small doubling: they are contained in a union of a polynomial number of affine subspaces of bounded size. Equivalent formulations include a robustness-to-linearity statement for functions f: F2n → F2n with many additive coincidences.

The conjecture would follow from a Polynomial Bogolyubov conjecture asserting large subspaces inside triple sumsets. Current best results (e.g., Sanders) obtain quasi-polynomial-type bounds, and special cases (e.g., monotone A) have been proved.

References

The PFR Conjecture is known to follow from the Polynomial Bogolyubov Conjecture [GT09a]: Let A ⊆ F2n have density at least α. Then A + A + A contains an affine subspace of codimension O(log(1/α)). One can slightly weaken the Polynomial Bogolyubov Conjecture by replacing A + A + A with kA for an integer k > 3. It is known that any such weakening (for fixed finite k) is enough to imply the PFR Conjecture.

— Open Problems in Analysis of Boolean Functions  (1204.6447 - O'Donnell, 2012) in Main matter, problem “Polynomial Freiman–Ruzsa Conjecture (in the F2^n setting),” remarks, first and second bullets