Congruence classes in the split finite-field case

Determine under what additional geometric conditions on a set E, including suitable non-concentration on isotropic lines, analogues of the triangle and higher-point congruence-class lower bounds hold when p≡1 mod 4.

Background

The main results assume p≡3 mod 4, which makes the quadratic form x_12+x_22 anisotropic and permits the rigid-motion incidence argument over the quadratic extension field. For p≡1 mod 4, isotropic lines exist, and sets contained in such a line can determine substantially fewer congruence classes than the bounds proved in the paper.

The question seeks geometric hypotheses—such as non-concentration on isotropic lines—that would exclude this obstruction and allow analogous lower bounds for triangles, four-point configurations, and fixed higher-point configurations in the split case.

References

Suppose that $p\equiv1\pmod4$. Under what additional geometric conditions on $E$, such as suitable non-concentration on isotropic lines, do analogues of Corollary~\ref{cor:eleven-sixths} and Theorem~\ref{thm:k-points} hold?

— Congruence Classes in $\mathbb{F}_p^2$: Sharp Results via an Energy Approach  (2609.29784 - Pham et al., 24 Sep 2026) in Section 5, “Open questions”