arXiv:2302.05240v5

Resonance between planar self-affine measures

Aleksi Pyörälä

math.CAmath.DS28A8037A10

Paper metadata

Original submission: February 10, 2023
Latest revision: August 7, 2026 · version 5
Claimed MSC: 28A80, 37A10
arXiv categories: math.CA, math.DS
Paper license: CC BY 4.0 ↗
Comments: 55 pages. Proof of Proposition 6.1 contained an error and its statement has been slightly modified. Main results unchanged
Journal reference: Advances in Mathematics, Volume 451, 2024, 109770, ISSN 0001-8708
DOI: 10.1016/j.aim.2024.109770 ↗

Abstract

We show that if $\lbrace \varphi_i\rbrace_{i\in Γ}$ and $\lbrace ψ_j\rbrace_{j\inΛ}$ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures $μ$ and $ν$ , the inequality $\dim_{\rm H}(μ*ν) < \min \lbrace 2, \dim_{\rm H} μ+ \dim_{\rm H} ν\rbrace$ implies that there is algebraic resonance between the eigenvalues of the linear parts of $\varphi_i$ and $ψ_j$ . This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.

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Audit summary

Audited against arXiv v5

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The resonance theorem, the general lower bound for convolution dimension, and the set-valued corollary are correct. The proof's varying-scale quantifier error does not disprove Theorem 1.2: its actual application admits a verified countable-restriction repair.

Theorem 1.1Correct

Resonance between dominated planar self-affine measures

Pages 3-4 and 30-31 · Theorem 1.1 and its proof · arXiv:2302.05240v5

Under irreducibility, domination, strong separation, full support, and $\dim\mu\geq\dim\nu$ , the proof correctly reduces any dimension drop to $\dim\mu>1>\dim\nu$ and then combines the fiber-structure estimate with the product-flow entropy estimate. If the logarithmic eigenvalue set is not arithmetic, a pair has irrational ratio; Proposition 6.1 supplies a common sampling time for which the relevant product flow is ergodic, and Claims 4.3-4.4 inserted into (4.4) give $\dim(\mu*\nu)\geq\min\{2,\dim\mu+\dim\nu\}$ , contradicting the assumed drop.

Full paper, version 5 ↗

Theorem 1.2Correct

Lower bound for convolution dimension

Page 4, pages 23-27, and pages 49-51 · Theorems 1.2 and 3.2; Claim 4.2 · arXiv:2302.05240v5

Claim 4.2 gives the required entropy lower bound on the increasing sets $X_m=\{(\mathbf i,\mathbf j):d(\theta(\mathbf i),\theta(\mathbf j))\geq1/m\}$ , whose union has full measure. For each fixed $m$ , a single sufficiently large $\beta_m$ works almost everywhere on $X_m$ . Apply the fixed- $\beta$ tree argument to the normalized restriction on $X_m$ ; differentiation makes its cylinder conditionals asymptotic to the unrestricted ones. If a Borel set has positive $\mu*\nu$ -measure, its preimage under addition meets some $X_m$ in positive measure, so the restricted lower bound forces that set to have dimension at least $\min\{1,\dim\mu\}+\min\{1,\dim\nu\}-\varepsilon$ . Letting $\varepsilon\downarrow0$ proves the theorem. This is a complete repair of the defective general varying- $\beta$ paragraph identified in Part 2.

Hochman-Shmerkin, local entropy averages ↗

Corollary 1.3Correct

Resonance statement for self-affine attractors

Page 5 · Corollary 1.3 and its proof · arXiv:2302.05240v5

The cited variational approximation produces fully supported block-system measures with dimensions arbitrarily close to those of the two attractors. The strict dimension gap persists after this approximation, Theorem 1.1 gives the arithmetic eigenvalue conclusion, and Theorem 1.2 excludes the remaining boundary case $\dim Y=1$ .

Full paper, version 5 ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The dependent- $\beta$ extension of the local-entropy theorem invokes Egorov's theorem to interchange an existential scale quantifier with a uniform tail quantifier, which is invalid. The actual use in Theorem 1.2 has a verified repair by the countable angle-separation decomposition. Three additional notation and scale-growth defects are harmless typos.

Proof of Theorem 3.2Incorrect as written

Egorov's theorem does not make a point-dependent scale uniform

Page 27 · dependent- $\beta$ paragraph after Equation (3.3) · arXiv:2302.05240v5

The hypothesis says that for almost every $(\mathbf i,\mathbf j)$ there exists some $\beta=\beta(\mathbf i,\mathbf j,\varepsilon)\geq N_0$ for which the entropy condition holds. The proof then claims that Egorov's theorem gives a large set $E$ and $N_1$ on which the condition holds for every $\beta\geq N_1$ . Egorov only makes convergence uniform for an already fixed sequence of functions; it cannot replace $\forall(\mathbf i,\mathbf j)\,\exists\beta$ by $\exists N_1\,\forall(\mathbf i,\mathbf j)\,\forall\beta\geq N_1$ . Thus the general extension is not proved. Downstream dependency: Claim 4.2 invokes precisely the point-dependent version in the proof of Theorem 1.2. Verified repair for that use: Claim 4.2 already partitions the full-measure set as $\bigcup_m X_m$ , and for each fixed $m$ a common $\beta_m$ works on $X_m$ ; apply the fixed-scale restriction argument separately to each $X_m$ . Every positive-measure preimage meets some $X_m$ positively, so the resulting countable family of restricted estimates proves the lower Hausdorff-dimension bound for the full convolution. This repair does not validate the broader variable- $\beta$ statement without additional hypotheses.

Full paper, version 5 ↗

Notation 3.1Typo

The magnification of $\mu$ is defined using the wrong measure

Page 23 · second display in Notation 3.1 · arXiv:2302.05240v5

Printed: $\mu_{\mathbf i|i_k}:=S_{k\beta}^*T_{\Pi(\mathbf i)}\varphi_{\mathbf i|i_k}\nu$ . Correction: replace the final $\nu$ by $\mu$ . The symbol is used thereafter as the magnification of $\mu$ , and the corrected formula is written explicitly at the start of the proof of Claim 4.2. No argument changes.

Full paper, version 5 ↗

Proof of Theorem 3.2Typo

The coding-length increments have order $\beta$ , not $\log\beta$

Page 25 · paragraph defining the finite alphabets $A$ and $B$ · arXiv:2302.05240v5

The paper says that the block-length increments are bounded by constant multiples of $\log\beta$ . The dyadic increment is $\lfloor(k+1)\beta\rfloor-\lfloor k\beta\rfloor$ , hence is of order $\beta$ ; the stopping-time increment for a fixed contracting IFS has the same order. The unique correction is to replace $\log\beta$ by $\beta$ . Only finiteness for each fixed $\beta$ is used, so the proof is unaffected.

Full paper, version 5 ↗

Proof of Claim 4.3Typo

The definition of $F'$ omits the map $F$

Page 51 · first paragraph of the proof of Claim 4.3 · arXiv:2302.05240v5

Printed: $F':Z'_\Phi\to\mathcal P(\mathbb R^2)$ is defined by $F'(\mathbf i,\theta,t)=(\mathbf i,\theta,1,t)$ , whose right-hand side is a point of $Z_\Phi$ , not a probability measure. Correction: $F'(\mathbf i,\theta,t)=F(\mathbf i,\theta,1,t)$ . All subsequent expressions treat $F'$ as this measure-valued composition, so the correction is unique and harmless.

Full paper, version 5 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2302.05240v5
Authors listed: Aleksi Pyörälä
Audit date: August 18, 2026

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Resonance between planar self-affine measures

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Resonance between dominated planar self-affine measures

Lower bound for convolution dimension

Resonance statement for self-affine attractors

Egorov's theorem does not make a point-dependent scale uniform

The magnification of μ\muμ is defined using the wrong measure

The coding-length increments have order β\betaβ, not log⁡β\log\betalogβ

The definition of F′F'F′ omits the map FFF

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The magnification of $\mu$ is defined using the wrong measure

The coding-length increments have order $\beta$ , not $\log\beta$

The definition of $F'$ omits the map $F$