arXiv:2607.14740v1

Projections of self-affine sets onto lines

Balázs Bárány, Antti Käenmäki, István Kolossváry

math.CAmath.DS28A8028A7537C4537A0515A75

Paper metadata

Original submission: July 16, 2026
Claimed MSC: 28A80, 28A75, 37C45, 37A05, 15A75
arXiv categories: math.CA, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 69 pages, 1 figure

Abstract

We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in $\mathbb{R}^d$ . Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set $X$ itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if $X$ additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.

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Audit summary

Audited against arXiv v1

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

01Statements4 reported findingsCorrect

All four main projection results are supported. Theorems 1.1–1.2 give the all-direction dimension formula for self-affine measures, including the planar refinement. Theorem 1.3 transfers it to self-affine attractors through high-dimension Bernoulli subsystems. Theorem 1.4 verifies that the required pinching and exterior-power irreducibility conditions hold in the announced generic/Zariski-dense parameter regime.

Theorems 1.1–1.2Correct

All-direction projection dimensions for self-affine measures

Pages 4–5 · Theorems 1.1–1.2 · arXiv:2607.14740v1

For the self-affine measures satisfying the exact separation and the stated proximality/pinching and irreducibility hypotheses, every nonzero one-dimensional projection has dimension equal to the minimum of one and the measure dimension. Theorem 1.2 gives the planar refinement under its weaker specialized assumptions. The conclusion is uniform over all projection directions, including the exceptional directions not covered by the initial Furstenberg-measure almost-everywhere theorem. The entropy dimension used in the proof agrees with Hausdorff dimension under the separation assumptions.

Theorem 1.3Correct

All-direction projections of self-affine sets

Page 5 and Pages 51–59 · Theorem 1.3 · arXiv:2607.14740v1

The attractor contains strongly separated dominated subsystems carrying Bernoulli measures whose dimensions approach the attractor dimension. Applying Theorems 1.1–1.2 to each subsystem gives the lower bound for every projection direction. The Lipschitz upper bound is immediate. Letting subsystem dimension increase yields $\min\{1,\dim_{\mathrm H}K\}$ for every line projection, with the exact-separation and group hypotheses retained.

Theorem 1.4Correct

Generic algebraic hypotheses

Page 5 and Pages 59–65 · Theorem 1.4 · arXiv:2607.14740v1

The complement of simultaneous proximality and irreducibility in the required exterior powers is contained in proper algebraic subvarieties. A Zariski-dense tuple therefore escapes those obstructions, and small perturbations preserve the pinching inequalities and exact separation on the stated open set. This gives the genericity conclusion in precisely the parameter range advertised, rather than asserting it for all affine systems.

Fixed-direction conclusionCorrect

Removal of Furstenberg-exceptional directions

Pages 32–50 · Sections 4–5 · arXiv:2607.14740v1

For a fixed direction, long random matrix products enter a compact subset of good projective directions with arbitrarily high probability. The entropy comparison across that product has a uniform bounded error from distortion and separation. Dividing by scale and sending block length to infinity transfers the almost-everywhere Furstenberg estimate to the fixed direction. Since the argument begins with an arbitrary direction, it proves the all-direction statement.

02Proofs3 reported findingsCorrect

Local entropy averages, Zariski escape, and separated subsystems. The Furstenberg stationary measure first supplies the projection-dimension formula for almost every direction. Local entropy averages express a fixed projection's dimension through conditional entropies at successive matrix scales. Proximality and exterior-power irreducibility let typical products move any fixed direction into the full-measure good set with controlled distortion; exact separation prevents entropy loss at that move. Averaging those blocks removes all exceptional directions. For sets, strongly separated dominated subsystems approximate both the affinity and Hausdorff dimensions, allowing the measure result to pass to the attractor.

Entropy and subsystem argumentsCorrect and complete

Local entropy averages, Zariski escape, and separated subsystems

Pages 20–65 · Sections 3–6 · arXiv:2607.14740v1

The Furstenberg stationary measure first supplies the projection-dimension formula for almost every direction. Local entropy averages express a fixed projection's dimension through conditional entropies at successive matrix scales. Proximality and exterior-power irreducibility let typical products move any fixed direction into the full-measure good set with controlled distortion; exact separation prevents entropy loss at that move. Averaging those blocks removes all exceptional directions. For sets, strongly separated dominated subsystems approximate both the affinity and Hausdorff dimensions, allowing the measure result to pass to the attractor.

Sections 3–5Correct and complete

Local entropy averages and projective transport

Pages 20–51 · Sections 3–5 · arXiv:2607.14740v1

Dyadic/local entropy averages compute projected dimension from conditional Shannon entropies with an error tending to zero. Random matrix products are split at stopping times adapted to singular-value scales. Strong irreducibility and proximality give the needed projective mixing, while exact separation controls component overlap. The subadditive and stopping-time errors are integrable and vanish after normalization, yielding a uniform lower bound for each fixed direction.

Section 6Correct and complete

Separated subsystems and algebraic escape

Pages 51–65 · Section 6 · arXiv:2607.14740v1

Long words are selected with disjoint cylinders and nearly constant singular-value data, producing dominated Bernoulli subsystems whose dimensions approach the target. Exterior-power pinching and irreducibility pass to these subsystems after adjoining finitely many connector words. The algebraic escape lemma chooses those connectors outside every proper invariant subvariety. Applying the measure theorem and taking the subsystem limit proves the set and genericity results.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.14740v1
Authors listed: Balázs Bárány, Antti Käenmäki, István Kolossváry
Audit date: August 18, 2026

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P(delta)

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Projections of self-affine sets onto lines

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

All-direction projection dimensions for self-affine measures

All-direction projections of self-affine sets

Generic algebraic hypotheses

Removal of Furstenberg-exceptional directions

Local entropy averages, Zariski escape, and separated subsystems

Local entropy averages and projective transport

Separated subsystems and algebraic escape

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