Sharp 3-energy bound for Katz–Tao sets on the parabolic cylinder

Establish that every δ-separated (δ, d)-Katz–Tao set S contained in the truncated parabolic cylinder Y = {(x, x², z) : x ∈ [0, 1], z ∈ [0, 1]} ⊂ R³, with 1 < d ≤ 2, satisfies E₃,δ(S) ≲ε δ⁻ε max{δ⁻(3d+2), δ⁻(5d−1)} for every ε > 0.

Background

The paper proves a sharp 3-energy estimate for δ-separated Katz–Tao sets on the truncated cone in R³. The truncated parabolic cylinder is presented as a related zero-Gaussian-curvature model for the cone, and the authors formulate the same two competing exponents for its 3-energy.

The conjectured exponents arise from two explicit configurations: nearly planar strips of width approximately √δ and sets whose projection to the x-coordinate is an arithmetic progression. The paper explains that resolving the cylinder conjecture would in particular address the corresponding unresolved 3-energy problem for the parabola.