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Abstract:We prove exact Fourier-dimension formulas for scalar dyadic Mandelbrot cascades pushed forward to fixed nondegenerate $C^2$ embedded arcs and fixed nondegenerate $C^2$ Jordan curves in $\mathbb R^2$. Let $W$ be in the minimal Kahane--Peyriere regime. For each fixed nondegenerate $C^2$ embedded arc $\gamma:[0,1]\to\mathbb R^2$, the pushforward $\mu_\gamma$ of the interval cascade satisfies, almost surely on non-extinction, \[
\dim_{\mathrm F}(\mu_\gamma)=A_{\mathrm{loc}}(W), \] where \[
A_{\mathrm{loc}}(W)
=
\sup_{q>1}
\max\left\{
0,\,
\frac{q-1-\log_2\mathbb E[W^q]}{q}
\right\}, \] with the $q$-term interpreted as $0$ when $\mathbb E[W^q]=\infty$. The analogous formula holds for scalar circle cascades pushed forward by fixed nondegenerate $C^2$ Jordan curves $\gamma:\mathbb T\to\mathbb R^2$, with the pushforward denoted by $\mu_\gamma^{\mathbb T}$.
This extends the scalar circle endpoint formula from the canonical circle to fixed parametrized arcs and Jordan curves. The main new issue beyond the canonical circle is the loss of the explicit trigonometric phase and, for arcs, the presence of endpoint stationary regimes. We prove the arc lower bound by a finite-$r$ annular Fourier theorem based on an endpoint-safe phase decomposition, phase-bin coefficient estimates, predictable capping, complex Freedman concentration, and an $r$-tail compensator. The Jordan lower bound follows by first-generation dyadic cutting into two fixed arcs. The matching upper bounds use deterministic curved-support obstructions together with the scalar-circle minimum lower local-dimension theorem.

Submission history

From: Xiang Fang [view email]
[v1] Wed, 10 Jun 2026 07:32:27 UTC (33 KB)
[v2] Thu, 2 Jul 2026 05:32:51 UTC (40 KB)