Canonical Mandelbrot Cascades on Curves Are Rajchman
We settle the Rajchman problem for canonical scalar dyadic Mandelbrot cascades at the minimal Kahane--Peyrière integrability threshold. If $μ$ is the cascade on $[0,1]$, then $\widehatμ(ξ)\to 0$ as $|ξ|\to\infty$, almost surely on non-extinction. For every fixed nondegenerate $C^2$ embedded arc $γ:[0,1]\to\mathbb{R}^2$, the pushforward $γ_\#μ$ is likewise Rajchman almost surely on non-extinction. The analogous conclusion holds for the scalar cascade on the parameter circle pushed forward by any fixed nondegenerate $C^2$ Jordan curve. No moment condition of order strictly greater than one is imposed; in particular, the results include the regime $\mathbb{E}[W^q]=\infty$ for every $q>1$. The proof combines a spine-based lower-deviation principle, adaptive terminal approximation, and predictable capping to obtain almost-sure estimates uniform over large frequency annuli without higher moments. For curved pushforwards, an endpoint-safe phase decomposition controls direction-dependent stationary regions, including those meeting the endpoints of an arc, and couples the geometric and probabilistic arguments through a common dyadic kernel. Combined with the exact Fourier-dimension formulas for the corresponding models, the theorems show that Rajchman decay persists at zero Fourier dimension.