Fourier dimension of finite-type convex domains

Establish the conjectural formula for the Fourier dimension of Gaussian multiplicative chaos restricted to the finite-type convex domains D_m defined by |x'|^m+x_d^2<1, namely determine whether it equals min{D_{γ,d}, 2+α_{γ,d-1}, 2+(2(d-1)+γ^2)/m} almost surely.

Background

The paper analyzes Fourier dimension for domains with flat boundary patches and for uniformly strictly convex domains. The subsection “Further geometric questions” considers the intermediate finite-type geometry D_m={ (x',x_d): |x'|^m+x_d^2<1 }, whose curvature vanishes at two isolated points and whose supporting-hyperplane contact has finite order m.

A supporting-cap heuristic yields the displayed candidate minimum of the bulk multifractal obstruction, the uniformly curved boundary obstruction, and a new finite-type obstruction. The authors explain that proving this conjecture would require a surface estimate uniform in frequency when the stationary point approaches a point of vanishing curvature.