Exact Fourier dimensions of dyadic Mandelbrot cascades under minimal integrability

Abstract

We determine the Fourier dimension of dyadic Mandelbrot cascades under the minimal Kahane-Peyriere integrability condition. The interval theorem is proved in a vector-valued dyadic cascade model in which sibling weights may have arbitrary dependence. For every balanced energy-admissible vector law, almost surely on non-extinction, dim_F(mu)=dim_E(mu)=dim_2(mu)=D_E(X). In the canonical scalar case, under W>=0, E W=1, E[W log_2^+ W] 1} max{0, (q-1-log_2 E[W^q])/q}. The interval and circle formulas share a light-tail/heavy-tail dichotomy but have different mechanisms: energy dimension for the interval, and minimum lower local dimension for the circle. The circle lower bound follows from a finite-moment annular Fourier theorem.