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 $\R^3$, $1<d\le 2$, 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 $L^6$ moment estimate for the Fourier transform of Frostman measures supported on the cone.