Exact Fourier dimensions of dyadic Mandelbrot cascades on curves of nonvanishing curvature under minimal integrability

Abstract

We prove an exact Fourier-dimension formula for scalar dyadic Mandelbrot cascades pushed forward to fixed C^2 Jordan curves with nonvanishing curvature. Let W be in the minimal Kahane-Peyriere regime, let the scalar dyadic cascade live on T = R/Z, and let gamma map T to R^2 be a fixed C^2 Jordan curve with nonvanishing curvature, parametrized at constant speed. For the push-forward measure mu_gamma, we prove that, almost surely on non-extinction, its Fourier dimension is A_loc(W), the usual local exponent obtained by optimizing over q>1 from the moment expression involving E[W^q]. The upper bound follows from the scalar circle local-dimension theorem, bi-Lipschitz transfer to the fixed curve, and a deterministic curved-support obstruction for Fourier dimension. The lower bound follows from a fixed-curve finite-r annular theorem, which gives summable annular Fourier decay under a single finite moment witness. The main analytic input is a deterministic phase-geometry package for fixed nondegenerate C^2 curves: stationary tubes, derivative bands, and phase-bin coefficient estimates replacing the explicit trigonometric structure available on the unit circle.