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arxiv: 2606.11758 · v1 · pith:35277JKRnew · submitted 2026-06-10 · 🧮 math.PR

Exact Fourier dimensions of dyadic Mandelbrot cascades on curves of nonvanishing curvature under minimal integrability

Yin Cai , Xiang Fang , Hongdou Qu This is my paper

Pith reviewed 2026-06-27 08:40 UTC · model grok-4.3

classification 🧮 math.PR
keywords Fourier dimensionMandelbrot cascadesdyadic cascadespush-forward measuresJordan curvesnonvanishing curvaturefractal measuresannular decay
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The pith

The Fourier dimension of the push-forward of a dyadic Mandelbrot cascade onto a fixed C^2 Jordan curve equals the local moment exponent A_loc(W) almost surely on survival.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when a scalar dyadic Mandelbrot cascade on the circle is pushed forward by a fixed curve of nonvanishing curvature, the resulting random measure has Fourier dimension exactly equal to the local exponent obtained by optimizing the moment function E[W^q] over q>1. This holds almost surely whenever the cascade does not die out, and requires only the minimal integrability that defines the Kahane-Peyriere regime. The result extends the circle case by replacing explicit trigonometric cancellation with a deterministic geometric control package built from stationary tubes, derivative bands, and phase-bin estimates. A reader would care because the formula gives a computable, exact value for the Fourier dimension of these fractal measures on curved supports rather than merely bounds.

Core claim

For the push-forward measure mu_gamma, almost surely on non-extinction, its Fourier dimension is A_loc(W), the usual local exponent obtained by optimizing over q>1 from the moment expression involving E[W^q]. The upper bound follows from the scalar circle local-dimension theorem, bi-Lipschitz transfer to the fixed curve, and a deterministic curved-support obstruction for Fourier dimension. The lower bound follows from a fixed-curve finite-r annular theorem, which gives summable annular Fourier decay under a single finite moment witness. The main analytic input is a deterministic phase-geometry package for fixed nondegenerate C^2 curves: stationary tubes, derivative bands, and phase-bin coeff

What carries the argument

The deterministic phase-geometry package of stationary tubes, derivative bands, and phase-bin coefficient estimates for a fixed C^2 Jordan curve with nonvanishing curvature, which produces the annular Fourier decay needed for the lower bound.

Load-bearing premise

The curve must be C^2, Jordan, parametrized at constant speed, and have nonvanishing curvature so that the phase-geometry estimates can replace trigonometric cancellation.

What would settle it

A concrete counterexample consisting of one explicit C^2 curve with nonvanishing curvature and one weight distribution W satisfying the minimal moment condition, for which the Fourier dimension of the surviving push-forward measure differs from A_loc(W).

read the original abstract

We prove an exact Fourier-dimension formula for scalar dyadic Mandelbrot cascades pushed forward to fixed C^2 Jordan curves with nonvanishing curvature. Let W be in the minimal Kahane-Peyriere regime, let the scalar dyadic cascade live on T = R/Z, and let gamma map T to R^2 be a fixed C^2 Jordan curve with nonvanishing curvature, parametrized at constant speed. For the push-forward measure mu_gamma, we prove that, almost surely on non-extinction, its Fourier dimension is A_loc(W), the usual local exponent obtained by optimizing over q>1 from the moment expression involving E[W^q]. The upper bound follows from the scalar circle local-dimension theorem, bi-Lipschitz transfer to the fixed curve, and a deterministic curved-support obstruction for Fourier dimension. The lower bound follows from a fixed-curve finite-r annular theorem, which gives summable annular Fourier decay under a single finite moment witness. The main analytic input is a deterministic phase-geometry package for fixed nondegenerate C^2 curves: stationary tubes, derivative bands, and phase-bin coefficient estimates replacing the explicit trigonometric structure available on the unit circle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves an exact Fourier-dimension formula for the push-forward μ_γ of a scalar dyadic Mandelbrot cascade (with weights W in the minimal Kahane-Peyrière regime) from the circle onto a fixed C² Jordan curve γ with nonvanishing curvature, parametrized at constant speed. Almost surely on non-extinction, the Fourier dimension equals A_loc(W), the local exponent obtained by optimizing the moment function E[W^q] over q>1. The upper bound is transferred from the known scalar circle result via bi-Lipschitz embedding together with a deterministic curved-support obstruction; the lower bound follows from a fixed-curve annular Fourier-decay theorem that requires only a single finite moment witness. The central analytic contribution is a deterministic phase-geometry package (stationary tubes, derivative bands, phase-bin coefficient estimates) that replaces the trigonometric structure available on the circle.

Significance. If the result holds, it supplies the first exact Fourier-dimension statement for these cascade measures on non-flat supports under the minimal (one-moment) integrability condition. The self-contained deterministic phase-geometry package is a reusable technical tool that could apply to other measures supported on C² curves. The argument achieves parameter-free transfer of the scalar upper bound and matches the lower-bound witness exactly to the definition of A_loc(W), which is a clear strength.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'minimal Kahane-Peyrière regime' is invoked without a one-sentence reminder of the precise moment condition; adding this would improve immediate readability for readers outside the subfield.
  2. The statement that the annular theorem 'gives summable annular Fourier decay under a single finite moment witness' is clear in outline but would benefit from an explicit cross-reference to the precise moment hypothesis used in the phase-bin estimates (e.g., the section containing the definition of the witness).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation transfers the known scalar-circle local-dimension result (an external theorem) to the curve via bi-Lipschitz embedding plus a deterministic curved-support obstruction, then obtains the matching lower bound from an annular-decay theorem whose phase-geometry estimates (stationary tubes, derivative bands, phase-bin coefficients) are supplied by a self-contained deterministic package that depends only on the fixed C^2 nonvanishing-curvature hypothesis and does not invoke or redefine A_loc(W). The single-moment witness used for annular sums is chosen to be consistent with the external definition of A_loc(W) via optimization over E[W^q], but the paper does not fit any parameter inside its own argument or rename a fitted quantity as a prediction. No self-citation is load-bearing for the central equality, and the deterministic estimates are independent of the moment expression.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the prior scalar circle theorem (standard in the field) and the new deterministic phase-geometry estimates for C^2 curves; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption C^2 regularity and nonvanishing curvature of the fixed Jordan curve gamma allow stationary tubes and derivative-band estimates that control phase alignment.
    Invoked to replace trigonometric identities available on the circle; stated in the abstract as the main analytic input.

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Reference graph

Works this paper leans on

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