曲线上的规范曼德勃罗级联是拉伊赫曼的
Canonical Mandelbrot Cascades on Curves Are Rajchman
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中文总结 AI 辅助
研究规范标量二元曼德勃罗级联的拉伊赫曼问题，通过结合多种方法，证明在特定条件下相关测度的傅里叶变换在无穷远处趋于零，且该结论对曲线推前情况也成立，揭示拉伊赫曼衰减在零傅里叶维数下持续存在。
中文摘要 AI 辅助
我们在最小的卡哈内 - 佩里耶可积性阈值下解决了规范标量二元曼德勃罗级联的拉伊赫曼问题。若\(\mu\)是\([0,1]\)上的级联，那么当\(|\xi|\to\infty\)时，\(\widehat{\mu}(\xi)\to 0\)几乎必然在非灭绝情况下成立。对于每个固定的非退化\(C^2\)嵌入弧\(\gamma:[0,1]\to\mathbb{R}^2\)，推前测度\(\gamma_\#\mu\)同样几乎必然在非灭绝情况下是拉伊赫曼的。类似结论对由任何固定非退化\(C^2\)约旦曲线推前的参数圆上的标量级联也成立。证明结合了基于脊柱的下偏差原理、自适应终端逼近和可预测封顶等方法。
英文摘要
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.
