3-energy conjecture for Katz–Tao sets on the parabola

Prove that every δ-separated (δ, s, C)-Katz–Tao set A contained in the parabola {(x, x²) : 0 ≤ x ≤ 1}, with 0 < s ≤ 1, satisfies E₃,δ(A) ≲ε,C,C∗ δ⁻ε max{δ⁻3s, δ⁻(5s−1)} for every ε > 0.

Background

The parabolic 3-energy problem is identified as a lower-dimensional analogue of the conjectured cylinder estimate. The paper explains that taking a product of a parabola set with a one-dimensional arithmetic progression would transfer the cylinder conjecture to the stated parabola bound.

Previous work cited in the paper gives partial improvements over elementary energy estimates, but the sharp discretized estimate remains unresolved. A Hausdorff-dimension formula for triple sumsets of regular subsets of the parabola is noted as insufficient by itself to imply this discretized energy conjecture.

References

Conjecture 1.3 (3-energy on the parabola). Let 0 < s ≤ 1, and let A ⊂ {(x, x2) : 0 ≤ x ≤ 1} be aδ-separated (δ, s, C)-KT set. Then for every ε > 0,E3,δ(A) ≲ε,C,C∗ δ−ε max{δ−3s, δ−(5s−1)}.

— Sharp $L^6$ Fourier moments and (3,$δ$)-energy estimates for fractal measures on the truncated cone in $\R^3$  (2610.23679 - Demeter, 16 Oct 2026) in Conjecture 1.3, Section 1.2, p. 3

We do not know a comparable implication from the cone estimate to the parabola estimate.

— Sharp $L^6$ Fourier moments and (3,$δ$)-energy estimates for fractal measures on the truncated cone in $\R^3$  (2610.23679 - Demeter, 16 Oct 2026) in Section 1.2, p. 4