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arxiv: 2404.15434 · v1 · submitted 2024-04-23 · 🧮 math.CA · math.PR

Fractal uncertainty principle for random Cantor sets

Xiaolong Han , Pouria Salekani This is my paper

Pith reviewed 2026-05-24 02:02 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords fractal uncertainty principlerandom Cantor setsFourier decayconcentration of measureharmonic analysisfractal measures
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The pith

With overwhelming probability, random Cantor sets in R satisfy the fractal uncertainty principle with exponent at least 1/2 - 3d/4.

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

The paper shows that Cantor sets of dimension d in (0, 2/3) built in the real line by a random procedure obey the fractal uncertainty principle at a stated exponent, with high probability. This continues earlier work on discrete random alphabets by moving to a continuous random construction. A reader would care because the principle gives explicit control on how much a function and its Fourier transform can both concentrate when their supports lie on fractal sets. The argument proceeds by proving a Fourier decay bound for the random measures that support the sets.

Core claim

We prove that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds for random Cantor sets with dimension d in (0,2/3) in R constructed via a different random procedure. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.

What carries the argument

Concentration of measure in the probability space of the random Cantor sets, which produces the Fourier decay estimate needed for the uncertainty principle.

If this is right

  • The Fourier transforms of the random measures decay at a rate sufficient to imply the stated FUP bound.
  • The result covers all dimensions d below 2/3 and uses a continuous random construction distinct from the discrete alphabet model.
  • The exceptional sets where the principle fails have probability that vanishes rapidly as the construction parameter grows.
  • The same concentration argument supplies the decay for the measures that generate the random sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar concentration techniques might extend the exponent range or apply to other randomly perturbed fractals.
  • The result indicates that averaging over random constructions can bypass obstacles that appear in deterministic fractal examples.
  • One could test whether the same probability space yields decay estimates strong enough for related problems such as restriction theorems on these sets.

Load-bearing premise

The chosen random construction of the Cantor sets yields a probability space in which concentration of measure applies directly to the Fourier transforms of the measures.

What would settle it

A sequence of explicit realizations of the random Cantor sets for which the Fourier transform of the associated measure decays slower than the rate required by the exponent 1/2 - 3d/4.

Figures

Figures reproduced from arXiv: 2404.15434 by Pouria Salekani, Xiaolong Han.

Figure 1
Figure 1. Figure 1: The initial three iterations and the alphabet used for a Cantor set in Ensemble I. (The intervals colored red are removed in the iteration process.) 0 1 2 A1 = {0, 1} 0 1 2 0 1 2 A2 = {0, 2} 0 1 2 0 1 2 0 1 2 0 1 2 A3 = {1, 2} [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The initial three iterations and the alphabets used for a Cantor set in Ensemble II. 0 1 2 {0, 1} 0 1 2 0 1 2 {0, 2}, {1, 2} 0 1 2 0 1 2 0 1 2 0 1 2 {1, 2}, {1, 2}, {0, 2}, {0, 1} [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The initial three iterations and the alphabets used for a Cantor set in Ensemble III. Remark (The FUP for Cantor sets in the discrete setting of ZN ). Let FN be the discrete Fourier transform which is unitary on l 2 (ZN ). For each j ∈ N, the discrete Cantor set Bj ∈ ZN . The FUP for Cantor sets in the discrete setting is concerned with the estimate of the form [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel (arXiv:2107.08276), we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension d is in (0,2/3). We proved that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds for these discrete Cantor sets with random alphabets. In this sequel, we construct random Cantor sets with dimension d in (0,2/3) in R via a different random procedure from the one in the prequel. We prove that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.

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 constructs random Cantor sets in R of dimension d in (0,2/3) via a new random procedure and proves that, with overwhelming probability, these sets satisfy the fractal uncertainty principle with exponent at least 1/2 - 3d/4 - ε for any ε>0. The argument proceeds by establishing a Fourier decay estimate for the associated random measures, relying on a concentration-of-measure phenomenon in the induced probability space.

Significance. If the result holds, it supplies the continuous-line counterpart to the authors' earlier discrete result (arXiv:2107.08276). The explicit use of concentration of measure to obtain the required Fourier decay for this particular random model is a technically clean contribution; the exponent is obtained without data-fitting or self-referential parameters.

minor comments (2)
  1. §2: the definition of the random procedure (the probability space on the sequences of intervals) would benefit from an explicit statement of the independence assumptions before the concentration argument is invoked.
  2. The comparison with the discrete prequel could include a short remark on why the same exponent 1/2-3d/4- carries over without additional loss.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is established via an explicit probabilistic argument: a Fourier decay estimate for the random measures is derived from concentration of measure in the probability space induced by the new random Cantor construction in R. This step is carried out directly for d in (0,2/3) and does not reduce to any fitted parameter, self-definition, or load-bearing self-citation. The prequel is referenced only for high-level strategy and the discrete case; the continuous-case derivation is self-contained and independent. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters or invented entities are described; the argument rests on standard real analysis, Fourier transforms, and probabilistic concentration inequalities.

axioms (2)
  • standard math Standard properties of Hausdorff dimension, self-similar measures, and Fourier transforms on R.
    Invoked throughout the construction and decay estimates.
  • domain assumption Concentration of measure holds in the probability space defined by the random Cantor construction.
    Central to obtaining the Fourier decay with high probability.

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