Ben Green's Open Problem 29

References:

• Ben Green's Open Problem 29
• [Gr12] Green, Ben. "What is... an approximate group." Notices Amer. Math. Soc 59.5 (2012): 655-656.
• [Br13] Breuillard, Emmanuel, Ben Green, and Terence Tao. "Small doubling in groups." Erdős Centennial. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. 129-151.
• [Sa10] Sanders, Tom. "On a nonabelian Balog–Szemerédi-type lemma." Journal of the Australian Mathematical Society 89.1 (2010): 127-132.
• [CrSi10] Croot, Ernie, and Olof Sisask. "A probabilistic technique for finding almost-periods of convolutions." Geometric and functional analysis 20.6 (2010): 1367-1396.

theorem Green29.green_29 : True ↔ ∃ (C : ℝ) (c : ℝ), 0 < C ∧ 0 < c ∧ ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] (K : ℝ) (A : Finset G), 1 ≤ K → IsApproximateSubgroup K ↑A → ∃ S ⊆ A, C * K ^ (-c) * ↑A.card ≤ ↑S.card ∧ S ^ 8 ⊆ A ^ 4

Suppose that $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$, $|S| \gg K^{-O(1)} |A|$, with $S^8 \subset A^4$?

theorem Green29.green_29.variant (K : ℝ) : 1 ≤ K → ∃ (c : ℝ), 0 < c ∧ ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] (A : Finset G), IsApproximateSubgroup K ↑A → ∃ (S : Finset G), c * ↑A.card ≤ ↑S.card ∧ S ^ 8 ⊆ A ^ 4

Such a conclusion is known with $|S| \gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10].