Polynomial dependence in the linear-group abelian-structure theorem

Prove that there exist polynomially bounded functions M_1,M_2:Z_{>0}→Z_{>0} such that, for every field F, every integer d≥1, and every K-approximate group A⊂GL_d(F), some abelian subgroup H≤GL_d(F) satisfies |A^2∩H|≥(2K)^{-M_1(d)}|A^2|^{1/M_2(d)}.

Background

Theorem 1.12 establishes an abelian subgroup with a quasi-polynomial dependence on the dimension d of the ambient general linear group. The conjecture asks whether both dimension-dependent losses can instead be bounded by polynomially growing functions.

The paper explains that the main obstacles include the log log d loss in the regular-element proposition and the limited use of affine subspaces in place of lower-dimensional varieties such as conjugacy classes.