The entropy-rate dimension formula for self-similar measures

The entropy-rate dimension formula for self-similar measures on the real line asserts that the Hausdorff dimension of such a measure equals the minimum of one and the ratio of the entropy rate of the random walk on the generating affine maps to the Lyapunov exponent, even when distinct compositions of the maps coincide (exact overlaps). A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims this formula for every finite family of contracting similarities of the line. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the manuscript's introduction:
  • Moran and Hutchinson gave the similarity-dimension formula under the open set condition. Whether the dimension can fall below its natural bound when distinct compositions never coincide is the exact overlaps problem, associated with Simon for sets. The entropy-rate formulation, which also covers exact overlaps, is stated as Conjecture 3 in a survey by Varjú.
  • Hochman showed that a dimension drop forces superexponential concentration of cylinder maps, which gives the formula under exponential separation. Breuillard and Varjú, and Varjú, treated Bernoulli convolutions, including full dimension for every transcendental parameter. Rapaport proved the formula without exact overlaps for algebraic contraction ratios, and Rapaport and Varjú, and Feng and Feng, treated further homogeneous and algebraic cases.
  • Baker and, independently, Bárány and Käenmäki constructed systems without exact overlaps whose distinct cylinders approach one another at arbitrarily fast superexponential rates, so no quantitative separation follows from the absence of exact overlaps.

The original result

• Claim. Theorem 1.1 of the manuscript states that for every finite family of maps x ↦ r_ix + t_i of the real line with 0 < |r_i| < 1 and every strictly positive probability vector, the self-similar measure has Hausdorff dimension min{1, h/χ}, where h is the entropy rate of the random composed maps and χ the Lyapunov exponent. No separation assumption is made: exact overlaps, repeated maps and unequal or negative ratios are allowed. The statement concerns dimension, not absolute continuity.
• Method, as described: an entropy estimate for finite laws that detects information hidden at arbitrarily fine scales through equal-weight two-point laws, combined with a disintegration of coding blocks by symbol counts and with translation averaging. The only external dimension theorem used is exact dimensionality, due to Feng and Hu.
• Source. At commit 1581d4c: The entropy-rate dimension formula for self-similar measures on the line.
• Formalization, as described by the repository. The scope note lean/docs/148.md states that the Lean formalization proves the formula for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one, with signed unequal ratios and exact overlaps allowed. It also gives two consequences without exact overlaps: the usual entropy-over-Lyapunov formula for a common positive ratio, and the dimension formula for self-similar sets. The Comparator challenges are SelfSimilar and SelfSimilarCorollaries.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 22 October 2026)
2026-10-07	S. Kittle and C. Kogler, The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line (arXiv:2610.10511v1)	Parallel result (independent)	The paper proves the exact overlaps conjecture and its generalized version in dimension one, the same dimension formula. It states that a proof was independently announced in the OpenAI collection as paper 148, and that its entropy inequality for sums of independent random variables is a quantitative strengthening of Lemma 3.2 of that manuscript, with the explicit constant 120m2; the rest of its proof uses different methods. It records that the OpenAI manuscript dates its proof to 24 September 2026 and became public on 6 October, and that the authors first found a proof on 27 September. Its machine contribution statement says that the language model GPT-6 Astra found the first proof presented to the second author.	Preprint, version 1.
2026-10-08	MathVet (GitHub; review of openai/math, tables at repository commit bb5fdb6, 18 October 2026)	Review (fidelity of the formal statements)	Rates this family "full" and records sandboxed Comparator passes for SelfSimilar and SelfSimilarCorollaries.	MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
2026-10-10	cert-machine, openai/math lane K: runs 38019899278 and 38040214623	Verification (two kernels)	Records that Lean's default kernel and the nanoda kernel accept both solutions, in 456 s and 526 s.	Repository.
2026-10-10 (latest commit)	ckkogler/kk26-supplementary-material (GitHub, commit e4aabb8)	Supplementary material of the parallel result	Five manuscripts credited to GPT-6 Astra and prompted by C. Kogler, and a conversation transcript from 27 September to 5 October 2026. According to the README, three of them led to the Kittle–Kogler paper and two explore extensions after it: a sharp logarithmic bound for entropy loss under addition, and self-similar measures with virtually solvable rotations.	The README states that inclusion does not establish the correctness of every claim and refers to the main paper for statements and proofs.

Notes

• The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.