Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman

Abstract: We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_φ<sup>{\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $φ$ of covariance $\mathbb{E}[φ(θ)φ(θ&#39;)] = \log\frac{1}{\lvert e<sup>{iθ}-e<sup>{iθ&#39;}\rvert}$. Then, almost surely, $$\widehat{M_φ<sup>{\mathrm{crit}}}(n)\longrightarrow0$$ 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.