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Math-2688 · mathematics · Dynamical systems

Dimension of a real self-similar attractor without exact overlaps

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a finite nonempty real similarity family $\phi _i(x)=r_i$ $x+t_i$ with $0<|r_i|<1$, assume distinct nonempty words define distinct composed maps. If K is its compact attractor and $s>=0$ solves $sum_i$ $|r_i|^s=1$, must $dim_H$ $K=min(1,s)$, without any effective separation or algebraicity assumption?

Short English statement

For a finite nonempty real similarity family $\phi _i(x)=r_i$ $x+t_i$ with $0<|r_i|<1$, assume distinct nonempty words define distinct composed maps. If K is its compact attractor and $s>=0$ solves $sum_i$ $|r_i|^s=1$, must $dim_H$ $K=min(1,s)$, without any effective separation or algebraicity assumption?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-2688-A1 · Literature report · Human + LLM · gpt-6 astra ·

Human contribution summary on Math-2688: Samuel Kittle, Constantin Kogler

Human Contribution: Samuel Kittle and Constantin Kogler

Related independent contribution: Kittle--Kogler $2026$

Paper and relation to this problem.
Samuel Kittle and Constantin Kogler, The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line, arXiv:2610.10511v1 (posted October 7, 2026), give an independent proof of the one-dimensional entropy-rate dimension formula. Their Corollary 1.2 settles exactly the attractor question recorded here: for a finite family $g_i(x)=r_i x+b_i$ with $0<|r_i|<1$ and no exact overlaps, if $K$ is the attractor and $s\geq 0$ satisfies $\sum_i|r_i|^s=1$, then \[ \boxed{\dim_H K=\min\{1,s\}.} \] This imposes no exponential separation or algebraicity hypothesis and allows unequal and orientation-reversing contraction ratios.
Stronger measure theorem (Theorem 1.1).
For a finitely supported probability law $\theta=\sum_i p_i\delta_{g_i}$ on contracting real similarities, let $\nu$ denote its stationary self-similar measure. Write \[ h_\theta=\lim_{n\to\infty}\frac{H(\theta^{*n})}{n}, \qquad \chi_\theta=-\sum_i p_i\log|r_i|>0, \] where $H(\theta^{*n})$ counts the entropy of the distinct composed affine maps, not merely symbolic words. Kittle--Kogler prove \[ \boxed{\dim_H\nu=\min\left\{1,\frac{h_\theta}{\chi_\theta}\right\}} \] with exact overlaps permitted. This has the same conclusion and scope as the main theorem of OpenAI's Result 148. If there are no exact overlaps, $h_\theta=H(p)$; choosing $p_i=|r_i|^s$ gives $H(p)=s\chi_\theta$, yielding the lower bound for $\dim_H K$, while the standard cylinder cover gives the matching upper bound. Thus neither contribution leaves a case of the recorded one-dimensional attractor problem unresolved by the other.
Additional result: entropy loss under addition (Theorem 1.3).
A distinct contribution of Kittle--Kogler is a quantitative inequality for arbitrary independent, finitely supported real random variables. For a compactly supported probability law $\rho$, put \[ V_\rho(r)=\frac{4}{r^3}\int_{\mathbb R} \inf_{c\in\mathbb R}\int_{[a,a+r)}(x-c)^2\,d\rho(x)\,da, \qquad \mathcal E(\rho)=\int_0^\infty V_\rho(r)\,\frac{dr}{r}. \] Here $\mathcal E$ is the variance energy, which accumulates normalized local variance across logarithmic scales. For independent finitely supported $X_1,\ldots,X_m$, their Theorem 1.3 gives \[ \boxed{ \sum_{j=1}^m H(X_j) -H\!\left(\sum_{j=1}^m X_j\right) \leq 30m\sum_{j=1}^m\mathcal E(\operatorname{Law}(X_j)). } \] The authors identify this as an explicit quantitative strengthening of the finite-law estimate underlying OpenAI's Lemma 3.2. In their comparison, it implies a strengthened version of that lemma with constant $C_m=120m^2$, whereas the original proof method yields a constant of order $O(m^3)$. These constants refer to the corresponding finite-law bounds and should not be confused with the $30m$ coefficient in Theorem 1.3 itself.
Proof strategies and attribution.
The arguments are different. OpenAI's Result 148 uses randomized nested partitions, fair two-point submeasures, conditioning on block symbol counts, and telescoping entropy gains at disjoint fine-scale windows. Kittle--Kogler prove their entropy-loss inequality by Poisson partitions and a Margulis--Russo differentiation argument, and deduce the dimension formula using Hochman's entropy theorem and a variance-summation method developed in earlier work of the authors. Kittle--Kogler explicitly acknowledge OpenAI's independently announced Result 148 and describe the finite-law estimates as the central new ingredients of both proofs.
Chronology, machine assistance, and formalization.
OpenAI dates Result 148 to September 24, 2026 and released it publicly on October 6, 2026. Kittle--Kogler report discovering their initial proof on September 27, 2026 and posted their paper on October 7, 2026. Their machine-contribution statement credits GPT-6 Astra with the initial proof and entropy inequality, and describes subsequent human rewriting and an alternative proof of the dimension deduction. They report an approximately 47,000-line Lean formalization, available at https://github.com/ckkogler/exact-overlaps-one-dim-lean. The OpenAI contribution separately links its own Lean scope and comparator files. Neither dimension theorem, by itself, asserts absolute continuity, and the existence of proof code should not be mistaken for an independent review by this site.
Sources for this comparison.

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Math-2688-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2688

The attractor corollary asserts $\dim_HK=\min(1,s_*)$ for arbitrary signed real contracting similarities with no exact overlaps, without separation or arithmetic assumptions. Its definition and all-word equivalence cover the recorded absence of collisions between distinct nonempty words. Repeated generators would be exact overlaps and are excluded from this corollary. The manuscript's broader measure theorem allows collisions, but that extra scope is not needed for the attractor assertion.

Dimension of a real self-similar attractor without exact overlaps

OpenAI's claim

For any finite real self-similar probability measure, its dimension is claimed to equal the smaller of one and map-walk entropy rate divided by Lyapunov exponent, even with exact overlaps and signed unequal contractions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Consider a nonempty finite indexed family $\Phi=(\varphi_i)$ consisting of similarities $\varphi_i(x)=r_ix+t_i$ of the real line, with arbitrary translations and $0<|r_i|<1$, and positive probability vector $p$. Let $\mu$ be its self-similar measure, $\chi=\sum_i p_i\log(1/|r_i|)$, and $h_{\mathrm{RW}}=\lim_{n\to\infty}H(G_n)/n$, where $G_n$ is the random complete affine composition of $n$ independent $p$-distributed generators. OpenAI claims \[ \dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}. \] No separation is imposed; repeated generators and exact overlaps are allowed. The attractor corollary assumes absence of exact overlaps (distinct finite words give distinct complete maps) and claims $\dim_HK_\Phi=\min\{1,s_*\}$, with $\sum_i|r_i|^{s_*}=1$. The measure assertion is stronger than that no-overlap set corollary.

Outline of the argument

Manuscript 1 (family 148, 2026-09-24): Finite-law comparisons expose entropy hidden within grid cells as fair two-point information at finer scales. Blocking and conditioning on symbol counts fixes the composed contractions while preserving map entropy; a hypothetical dimension loss would create too much fair-pair mass in a fine-scale band, contradicting the entropy estimates.

Formal verification

For family 148, the scope page documents the following selections. The selected dimension identity permits signed unequal contractions and exact coincidences. Further declarations treat homogeneous measures and overlap-free attractors; density and absolute continuity are not selected. The documented statement covers the main claim of manuscript 1. Comparator files: SelfSimilar.lean, SelfSimilarCorollaries.lean. Declarations: OAI.EntropyRateDimension.entropy_rate_dimension, OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct, OAI.CorSetReference.cor_set.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

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Reviews 1

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