Supersaturation for finite-field linear configurations

Determine, for which F_q-linear configurations F and constants δ>0 satisfying ex_q(n,F)=O(N^{1−δ}), every linear configuration A⊆F_q^n of size αN≥CN^{1−δ} contains at least cα^{|F|}N^{rank(F)} linear copies of F.

Background

The paper defines a Sidorenko property for linear systems and explains that it is equivalent to a suitable supersaturation statement. The question asks for a characterization of those configurations for which an extremal bound at exponent 1−δ automatically yields the expected number of copies above the same density threshold.

References

For which $\mathbb F_q$-linear configurations $F$ and constants $\delta > 0$ satisfying $ex_q(n,F) = O(N{1-\delta})$ do there exist constants $C,c > 0$ such that every linear configuration $A \subseteq \mathbb F_qn$ of size $\alpha N \geq CN{1-\delta}$ contains at least $c \cdot \alpha{|F|}N{\mathrm{rank}(F)}$ linear copies of $F$, where $N := qn$?

— Generic solutions to symmetric linear equations  (2610.02177 - Frederickson et al., 1 Oct 2026) in Question 1, Section 1, subsection “Systems of linear equations”

For every integer $k \geq 2$ and prime power $q$, does there exist a constant $C$ such that for every linear configuration $A \subseteq \mathbb F_qn$ of size at least $CN{1/k}$, there exists an $\mathbb F_q$-linearly generic solution $(x_1, \ldots, x_{2k}) \in A{2k}$ to the equation x_1 + \cdots + x_k = x_{k+1} + \cdots + x_{2k},\tag{\ref{eq:equal k-sums} 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”