Improved regular-element structure in approximate linear groups

Prove that there exists a constant c>0 such that, whenever every element a∈A^2 of a K-approximate group A⊂GL_d(F) has an eigenspace of dimension at most (1−c)d, there are subspaces V_1⊂V_2⊂F^d with dim V_2−dim V_1>cd for which at least (2K)^{-O(d^{O(1)})}|A^2| elements a∈A^2 satisfy (a−λ)V_2⊂V_1 for some scalar λ∈F.

Background

Proposition 4.6 gives a weaker dichotomy involving either many not-too-irregular elements or substantial control by a well-behaved subgroup, with a loss involving log log d. This conjecture removes that logarithmic factor and formulates a stronger structural conclusion under a uniform eigenspace hypothesis.

The conjectured statement is intended to improve the quantitative dependence in the paper’s theorem on approximate subgroups of general linear groups.