Sharpness of Theorem 1.3 for every exponent

Determine whether Theorem 1.3 is sharp for every specific exponent α ∈ [0, 1], meaning whether for each such α there exist sufficiently large finite fields and subsets E ⊂ F_q^2 with |E| approximately q^α, |R_E| approximately q^{3α/2}, and E not contained in a line.

Background

Theorem 1.3 proves that if a subset E of the affine plane F_q2 has size at most a constant multiple of qα and is preserved by at least a constant multiple of qβ elements of SL_2(F_q), with β ≥ 3α/2, then E must be contained in a line. Example 5 establishes sharpness at exponents of the form α = 1/s or α = 2/s for positive integers s, using subfield examples.

The authors explicitly conjecture that this sharpness phenomenon extends to every α in the interval [0,1]. Establishing the conjecture would show that the exponent 3α/2 cannot generally be improved for any exponent in the stated range.

References

Based on Example 5, we conjecture that Theorem 1.3 is sharp for each specific α ∈ [0, 1].

— Sets preserved by a large subgroup of the special linear group  (2501.01697 - Hung et al., 3 Jan 2025) in Page 2, paragraph following Example 5

How far can the ranges in Corollary~\ref{cor:eleven-sixths} and Theorem~\ref{thm:k-points} be extended? In particular, what lower bounds hold when $$ |E|>p{3/4}\quad\text{for }k=3,4, $$ or when $$ |E|>p{3(k-2)/(3k-4)}\quad\text{for }k\geq5?$?

— 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”