AI 中文总结
该研究证明圆周上典范临界高斯乘性混沌的傅里叶系数几乎必然在无穷远处趋于零，解决了临界Rajchman问题，通过贝塞尔回归、条件浓度等技术克服了重尾质量与多频率控制的难点。
AI 中文摘要
我们证明了圆周上典范临界高斯乘性混沌的傅里叶系数在无穷远处几乎必然趋于零。更确切地说，设$M_ϕ^{\text{crit}}$是与中心化圆周场$ϕ$相关的典范临界混沌，该场的协方差为$\boldsymbol{E}[ϕ(θ)ϕ(θ')] = \frac{1}{\rvert e^{iθ}-e^{iθ'}\rvert}$的对数。那么，几乎必然地，当$|n|\to\to\to$时，$\to{M_ϕ^{\text{crit}}}(n)\to0$。这解决了典范圆周场的几乎必然临界Rajchman问题。由于临界混沌的傅里叶维数几乎必然为零，因此不存在正的多项式傅里叶衰减速率；因此，该定理展示了超出正傅里叶维数范畴的定性傅里叶抵消现象。\n证明过程解决了两个相互耦合的难点：临界混沌的重尾、非均匀单元质量，以及需要控制每个二进环域中指数级数量的频率。对于一个辅助的周期化紧支集星尺度场，基于导数的贝塞尔回归给出了加权小单元可和性以及异常大单元的移动尾控制。在傅里叶尺度以下的粗尺度上进行条件化后，有限程独立性和条件伯恩斯坦浓度将每个二进环域上终端傅里叶系数的一致控制简化为粗可预测测度的空间变分估计。随后，通过光滑正定协方差修正和临界混沌唯一性，将Rajchman性质转移到精确圆周场的典范临界混沌上。
英文摘要
We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_ϕ^{\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $ϕ$ of covariance $\mathbb{E}[ϕ(θ)ϕ(θ')] = \log\frac{1}{\lvert e^{iθ}-e^{iθ'}\rvert}$. Then, almost surely, $\widehat{M_ϕ^{\mathrm{crit}}}(n)\longrightarrow0$ as $\lvert n\rvert\to\infty$. This resolves the almost-sure critical Rajchman problem for the canonical circle field. Since critical chaos has Fourier dimension zero almost surely, no positive polynomial Fourier-decay rate can hold; the theorem therefore exhibits qualitative Fourier cancellation beyond the regime of positive Fourier dimension. The proof addresses two coupled difficulties: the heavy, nonuniform cell masses of critical chaos and the need to control exponentially many frequencies in each dyadic annulus. For an auxiliary periodized compact-range star-scale field, a derivative-rooted Bessel regression yields weighted small-cell summability and moving-tail control of exceptional large cells. After conditioning at a coarse scale below the Fourier scale, finite-range independence and conditional Bernstein concentration reduce uniform control of the terminal Fourier coefficients over each dyadic annulus to a spatial-variation estimate for a coarse predictable measure. A smooth positive-definite covariance correction and critical-chaos uniqueness then transfer the Rajchman property to the canonical critical chaos of the exact circle field.
Comments35 pages
