Improved bounds for odd-variable affine-type equations

Determine whether, for every prime power q≥3, there exists a function δ_q(k) tending to 1 as k→∞ such that every linear configuration in F_q^n avoiding linearly generic solutions to all affine-type equations with 2k+1 variables and nonzero coefficients has size O(N^{1−δ_q(k)}).

Background

For odd-variable affine-type equations over fields of characteristic at least 3, the paper cites temperateness results giving an exponent δ_q(k)>0, but notes that the known iterative slice-rank argument makes δ_q(k) shrink superexponentially with k. The question asks whether the exponent can instead approach 1 as the number of variables grows.

References

For every prime power $q \geq 3$, does there exist a function $\delta = \delta_q(k)$ tending to $1$ as $k \to \infty$ such that every linear configuration $A \subseteq \mathbb F_qn$ with no $\mathbb F_q$-linearly generic solution to any affine-type $\mathbb F_q$-linear equation \lambda_1x_1 + \cdots + \lambda_{2k+1}x_{2k+1}=0 with $2k+1$ variables and nonzero coefficients has size $O(N{1-\delta})$, where $N := qn$?

— Generic solutions to symmetric linear equations  (2610.02177 - Frederickson et al., 1 Oct 2026) in Question, Section 6, subsection “Linearly generic solutions to a single equation”