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The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture)

Let φi(x)=rix+ti\varphi_i(x)=r_ix+t_i (0<∣ri∣<10<|r_i|<1) be finitely many similarities of R\mathbb R and pp a positive probability vector, with self-similar measure μ=∑ipi(φi)∗μ\mu=\sum_ip_i(\varphi_i)_*\mu. Always dim⁡Hμ≤min⁡{1,hRW/χ}\dim_H\mu\le\min\{1,h_{\mathrm{RW}}/\chi\}, where hRW=lim⁡H(Gn)/nh_{\mathrm{RW}}=\lim H(G_n)/n is the entropy rate of the random composed map GnG_n and χ=−∑pilog⁡∣ri∣\chi=-\sum p_i\log|r_i|. The exact overlaps conjecture, associated with Simon (1996) and central since Hochman's 2014 work, asks whether equality dim⁡Hμ=min⁡{1,H(p)/χ}\dim_H\mu=\min\{1,H(p)/\chi\} holds whenever no two distinct words of equal length give the same map. The entropy-rate dimension conjecture (Varju's survey, Conjecture 3) extends this to systems with exact overlaps. Does dim⁡Hμ=min⁡{1,hRW/χ}\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\} hold for every finite self-similar system on the line?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Fractal geometry; dimension of self-similar measures
Posed by
Karoly Simon (exact overlaps question, 1996); entropy-rate form as Conjecture 3 in Peter Varju's survey (arXiv:2509.22042)
Year posed
1996
Years open
30y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every finite system of contracting similarities of R\mathbb R and every positive probability vector, dim⁡Hμ=min⁡{1,hRW/χ}\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}, with no separation or arithmetic assumption; exact overlaps, repeated maps, negative and unequal ratios are allowed. Without exact overlaps this gives min⁡{1,H(p)/χ}\min\{1,H(p)/\chi\} (the exact overlaps conjecture in dimension one) and the attractor formula dim⁡HK=min⁡{1,s}\dim_HK=\min\{1,s\} with ∑∣ri∣s=1\sum|r_i|^s=1. It says nothing about absolute continuity (for instance of Bernoulli convolutions) and does not treat higher-dimensional or non-conformal systems.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof uses Feng-Hu exact dimensionality as its only external dimension theorem.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against both formulations. It states dim⁡Hμ=min⁡{1,hRW/χ}\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\} for every finite indexed family with 0<∣ri∣<10<|r_i|<1 and every strictly positive weight vector, signed and unequal ratios and exact overlaps allowed; with no exact overlaps H(Gn)=nH(p)H(G_n)=nH(p), giving the exact overlaps conjecture on the line. Lean: lean/formalization.yaml lists comparator SelfSimilar with declaration OAI.EntropyRateDimension.entropy_rate_dimension. The statement ComparatorChallenges/SelfSimilar.lean was read here: for a finite system with nonzero ratios of modulus below one and positive weights summing to one, and any probability measure satisfying the self-similarity equation, the lower Hausdorff dimension (infimum of dimH over positive-measure Borel sets) equals min(1, entropy rate / Lyapunov exponent), both in base 2, the entropy rate defined as the infimum of H(Gn)/nH(G_{n})/n over complete affine maps. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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