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arxiv: 2606.23160 · v1 · pith:L2PSNPB7new · submitted 2026-06-22 · 🧮 math.CA · math.AP· math.DS· math.SP

Anisotropic 2D FUP and quantum open baker's map

Long Jin , An Zhang , Hong Zhang This is my paper

Pith reviewed 2026-06-26 06:22 UTC · model grok-4.3

classification 🧮 math.CA math.APmath.DSmath.SP
keywords fractal uncertainty principlequantum open baker's mapessential spectral gapBedford-McMullen carpetanisotropic fractalsporosity condition
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The pith

An anisotropic discrete fractal uncertainty principle implies an essential spectral gap for the 2D quantum open baker's map.

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

The paper establishes an essential spectral gap for the 2D anisotropic quantum open baker's map by proving a corresponding anisotropic discrete fractal uncertainty principle tied to the Bedford-McMullen carpet. This extends earlier one-dimensional results and two-dimensional isotropic cases. The work also derives a continuous fractal uncertainty principle for two-dimensional anisotropic porous sets under a newly introduced anisotropic line porosity condition. A reader would care because the gap controls the decay rate of quantum states in open chaotic systems, offering a concrete handle on how anisotropy shapes spectral behavior in higher dimensions.

Core claim

The authors prove an essential spectral gap for the 2D anisotropic quantum open baker's map. The central step is an anisotropic discrete fractal uncertainty principle for the Bedford-McMullen carpet. They further establish the continuous fractal uncertainty principle for two-dimensional anisotropic porous sets and introduce the anisotropic line porosity condition as the key hypothesis.

What carries the argument

The anisotropic discrete fractal uncertainty principle for the Bedford-McMullen carpet, which supplies the decay estimate needed to obtain the spectral gap.

If this is right

  • The essential spectral gap extends directly from the one-dimensional and isotropic two-dimensional settings to the anisotropic two-dimensional case.
  • The continuous fractal uncertainty principle holds for any two-dimensional anisotropic porous set obeying the new line porosity condition.
  • The relation between the discrete and continuous versions of the fractal uncertainty principle follows the pattern established in prior one- and two-dimensional work.
  • The spectral gap controls the rate at which quantum states escape in the open anisotropic baker's map.

Where Pith is reading between the lines

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

  • The same porosity condition may allow the discrete FUP to be verified for other self-affine carpets beyond the Bedford-McMullen example.
  • If the line porosity condition can be checked algorithmically, it would give a practical test for when anisotropic maps admit spectral gaps.
  • The result suggests that anisotropy does not destroy the gap phenomenon once the appropriate porosity is verified, which could guide extensions to non-rectangular domains.

Load-bearing premise

The Bedford-McMullen carpet and similar anisotropic fractals satisfy the anisotropic line porosity condition that makes the discrete fractal uncertainty principle hold.

What would settle it

Numerical computation of the quantum open baker's map spectrum showing that the gap size does not remain bounded away from zero as the discretization parameter tends to infinity, or an explicit counterexample to the anisotropic discrete FUP on the Bedford-McMullen carpet.

Figures

Figures reproduced from arXiv: 2606.23160 by An Zhang, Hong Zhang, Long Jin.

Figure 1
Figure 1. Figure 1: Anisotropic Cantor iterates, to a Bedford-Mcmullen carpet. Here we choose M1 = 3, M2 = 2 and A = {(0, 1),(1, 0),(2, 1)}. Let {Xk}, {Yk} ⊂ ZN be two sequences of discrete Cantor sets, and let X, Y ⊂ T 2 be their respective limiting Cantor sets, all constructed from a common base M and from two alphabets A and B (respectively) via (1.3). The following theorem, our second main result, completely characterizes… view at source ↗
Figure 2
Figure 2. Figure 2: Intersection of an oblique line with the anisotropic limiting Cantor set X for parameters M = (4, 3) and A = ZM \ {(1, 1)}. Proposition 3.13 (Line containment). Suppose the alphabet A ⊂ ZM on base M is not full. If some open subset of a line Rv + p lies entirely in the limiting set X, then [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The intersection of Φk (In0,k) and Φk (X). The key point is that the line has very small but positive slope. 3.4. Proof of Theorem 1.2. We now have all the ingredients to prove the anisotropic discrete FUP (1.4) for the discrete Cantor sets Xk and Yk defined in (1.3). Recall that Xk =  (a (0) 1 + · · · + a (k−1) 1 Mk−1 1 , a (0) 2 + · · · + a (k−1) 2 Mk−1 2 ) : (a (j) 1 , a (j) 2 ) ∈ A ⊂ ZN, Xk =  (x, y)… view at source ↗
Figure 4
Figure 4. Figure 4: Anisotropic porosity for Bedford-McMullen carpets with M1 = 4, M2 = 3 and A = ZM1 × ZM2 \ {(3, 0),(1, 1),(2, 2)}. The whole blue segment I is of scale ∼ 4 −2 with a sub-segment I ′ lying in the green rectangle of scale ∼ 4 −3 × 3 −3 . Example 4.9 (Anisotropic porous BM carpet). Let X ⊂ R 2 be the Bedford-McMullen carpet iterated from base M and alphabet A ⊂ ZM, where M2 = Mα 1 for some α ∈ (0, 1). Assume t… view at source ↗
Figure 5
Figure 5. Figure 5: Failure of Cohen’s line porosity for the Bedford-McMullen carpets with M1 = 4, M2 = 3 and A = {(1, 0),(2, 1),(0, 2)}. The black rectangle of size 4−3 × 3 −3 in the right picture has holes after further iteration, but after rescaling it looks identical to the original carpet. Any line segment of length 4−3 parallel to the x-axis, with its left endpoint on the left boundary of this sub-rectangle, has non-emp… view at source ↗
Figure 6
Figure 6. Figure 6: □ 4−3 × 3−3 scale [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
read the original abstract

We prove an essential spectral gap for 2D anisotropic quantum open baker's map. This extends the 1D results of Dyatlov--Jin 2017 and the isotropic 2D results of Cohen 2025a. The key ingredient is the anisotropic discrete fractal uncertainty principle (FUP) associated with a 2D anisotropic fractal set called the Bedford--McMullen carpet. We also study the relation between our anisotropic discrete FUP and its continuous counterpart in the spirit of Dyatlov--Jin 2018 and Cohen 2025a. In particular, we prove {continuous FUP} for 2D {anisotropic porous} sets, extending the (high-dimensional) isotropic results of Cohen 2025b. To the best of our knowledge, the anisotropic (line) porosity condition -- a variant of Cohen's line porosity and stronger than ball porosity -- appears to be new to the literature.

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 / 3 minor

Summary. The manuscript proves an essential spectral gap for the 2D anisotropic quantum open baker's map, extending the 1D results of Dyatlov-Jin (2017) and the isotropic 2D results of Cohen (2025a). The key technical step is an anisotropic discrete fractal uncertainty principle (FUP) for the Bedford-McMullen carpet. The paper also establishes a continuous FUP for 2D anisotropic porous sets and introduces a new anisotropic (line) porosity condition, which is claimed to be novel and stronger than ball porosity.

Significance. If the proofs hold, the work provides a meaningful extension of fractal uncertainty principles and spectral gap results into the anisotropic 2D regime. The introduction of the line porosity condition is a concrete contribution that may find use in other fractal geometry contexts. The manuscript explicitly builds on independent prior works (Dyatlov-Jin 2017/2018, Cohen 2025a/b) and studies the discrete-to-continuous relation without apparent circularity or free parameters.

minor comments (3)
  1. [Abstract] The abstract contains placeholder-style braces around terms such as {continuous FUP} and {anisotropic porous}; these should be removed or clarified in the final version.
  2. [Section 1] Notation for the anisotropy parameters and the precise statement of the line porosity condition should be introduced with a dedicated definition early in the paper to improve readability for readers familiar with Cohen's isotropic line porosity.
  3. [Section 4] The relation between the new anisotropic discrete FUP and the continuous counterpart is described as 'studied'; a brief comparison table or explicit statement of which implications are new versus recovered would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation to co-author's prior work, not load-bearing for new anisotropic result

full rationale

The paper extends 1D results from Dyatlov-Jin 2017 (overlapping author) and isotropic 2D results from Cohen 2025a, but the central claims rest on a newly introduced anisotropic line porosity condition for the Bedford-McMullen carpet and the associated discrete FUP proof. No equations or derivations reduce by construction to fitted inputs, self-definitions, or unverified self-citations; the new porosity variant is explicitly presented as appearing new to the literature. The self-citation provides context for the extension but does not justify the core anisotropic contribution, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of the anisotropic discrete FUP for the Bedford-McMullen carpet and the validity of the new anisotropic line porosity condition; these are presented as extensions of prior definitions rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Properties of the Bedford-McMullen carpet and standard fractal constructions from prior literature.
    Invoked as the fractal set for which the anisotropic FUP holds.
  • standard math Standard results from microlocal analysis and spectral theory for quantum maps.
    Background for the essential spectral gap statement.

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discussion (0)

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