Marton's conjecture (the polynomial Freiman-Ruzsa conjecture)

Aaron Anderson, Mantas Bakšys, Jonas Bayer, Mauricio Collares, Rémy Degenne, Yaël Dillies, Ben Eltschig, Sébastien Gouëzel, Kalle Kytölä, Rob Lewis, Paul Lezeau, Lorenzo Luccioli, Heather Macbeth, Patrick Massot, Arend Mellendijk, Kyle Miller, Pietro Monticone, Kim Morrison, Oliver Nash, Utensil Song, Terence Tao, Floris van Doorn, Sky Wilshaw, Lawrence Wu

Palomar Registrypalomar/PALOMAR-2026-09-06-000006Sep 6, 2026math.COscore 0

A Lean formalization of Marton's conjecture, widely known as the polynomial Freiman-Ruzsa conjecture, together with its principal consequences. If A is a non-empty subset of an abelian group of exponent 2 with |A + A| = 2, the covering number is m * K^(256m^3 + 1). Three consequences are also recorded: a finite set of small doubling in Z^D has a subset of at least K^(-34) of its size whose affine dimension is at most (80/log 2) log K; a map between finite groups of exponent 2 whose additive defects f(x+y) - f(x) - f(y) all lie in a set S differs from a homomorphism by a map taking at most…

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