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_q^2 has size at most a constant multiple of q^α and is preserved by at least a constant multiple of q^β 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.