Sharp $L^6$ Fourier moments and (3,$δ$)-energy estimates for fractal measures on the truncated cone in $\R^3$

Abstract: For a $δ$-separated $(δ,d)$-Katz--Tao set $P$ on the truncated cone in $\R<sup>3$, $1&lt;d\le 2$, we prove the sharp energy bound [ \E_{3,δ}(P)\lesssim_{\varepsilon} δ^{-\varepsilon} \max{δ^{-(3d+2)},δ^{-(5d-1)}}. ] The proof applies the Fu--Ren incidence theorem to triples with a fixed sum. As a corollary we also deduce a sharp $L<sup>6$ moment estimate for the Fourier transform of Frostman measures supported on the cone.