2026/07/17 by Yin Cai, Guozheng Cheng, Xiang Fang +3
#math.PR #math.CA
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)→ 0 as |ξ|→∞, almost surely on non-extinction. For every fixed nondegenerate C2 embedded arc γ:[0,1]→ℝ2, 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 C2 Jordan curve. No moment condition of order strictly greater than one is imposed; in particular, the results include the regime 𝔼[Wq]=∞ 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.