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The geometric case of the Erdős similarity conjecture
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Abstract
We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed $q\in(0,1)$ and every $\eta\in(0,1)$, we construct a compact set $E_{q,\eta}\subseteq[0,1]$ of measure greater than $1-\eta$ containing no nontrivial affine copy of $\{q^n:n\ge1\}$, for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Kan Jiang and Lifeng Xi, The Erdős similarity problem for null sequences with adjacent ratios bounded below (arXiv:2610.11449, 8 October 2026). Builds compact sets of measure above $1-\eta$ in $[0,1]$ that avoid affine copies of positive null sequences with $\liminf a_{n+1}/a_n>0$ (without monotonicity, and simultaneously for prescribed countable families), adapting the finite-tree construction of the dyadic and geometric manuscripts, which it cites as unpublished manuscripts [17, 18]. [family-level]
- Vjekoslav Kovač (submitter), generated with ChatGPT-6 Astra, Superexponentially decaying sequences in the Erdos similarity problem (hexagon:2610.00176v1, 10 October 2026). Proves a non-universality criterion that covers some superexponentially decaying sequences, and states that its proof modifies the finite-tree construction of this geometric-case manuscript (its reference [18]). [AI-generated per source]
- Harald Schilly, Erdős similarity for distance sets: finite patterns, a pinned obstruction, and an open window (preprint and Lean repository, October 2026). This follow-up, prepared with substantial Claude assistance, derives pinned-distance obstructions from the avoidance results and reports replaying the dyadic Lean proof, while retaining general-ratio avoidance as an explicit hypothesis. [family-level]
Formalization
- OpenAI math release, Lean scope note for family 084 (release Lean documentation). The scope note covers only the dyadic case $q=1/2$ of the companion paper (compact subsets of $[0,1]$ of measure arbitrarily close to one avoiding every translated dyadic sequence with either sign of nonzero dilation); there is no formalization of the general-$q$ theorem. [family-level]
- OpenAI math release, DyadicAvoidance/Main.lean (openai/math, commit adc7f12). Lean theorem
Problem310.dyadic_affine_avoidance, the $q=1/2$ special case of Theorem 1.1. [family-level]
Related prior literature mentioned by others
- Yeonwook Jung, Chun-Kit Lai, Yuveshen Mooroogen, Fifty years of the Erdős similarity conjecture (arXiv:2412.11062v2, 2025-01-01). This survey states the conjecture "remains open for exponentially decaying sequences" and reviews variants of measure universality; the Jiang–Xi, Hexagon and Schilly follow-ups cite it, and it is not in this paper's bibliography.
Possibly relevant literature (found by the agent, not checked)
- Alex Burgin, Samuel Goldberg, Tamás Keleti, Connor MacMahon, Xianzhi Wang, Large sets avoiding infinite arithmetic / geometric progressions (arXiv:2210.09284, October 2022). Builds a compact set with a density point at 0 that contains no non-constant infinite geometric progression of any ratio, a variant of the fixed-ratio avoidance problem in Theorem 1.1; not in this paper's bibliography.
Digests and expositions
- Patrick White with Claude, Erdős Problem a Day, Erdős #120: OpenAI claims the geometric case (2026-10-14). Explains the routing and scale-counting construction and reports AI-led proof audits and a dyadic Lean replay, explicitly distinguishing these from human verification of the general-ratio theorem. [AI-generated per source] [family-level]
Elsewhere
- Hexagon: Superexponentially decaying sequences in the Erdos similarity problem (AI-generated; human understanding: some)
- VibeMathed: Problem record
- Emergent Mind: family 084
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026,
author = {{OpenAI}},
title = {{The geometric case of the Erd\H{o}s similarity conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf}{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/084.md. The Lean statements have not been checked by people against the paper's claims.
Family 084: The geometric case of the Erdős similarity conjecture
For every fixed $q\in(0,1)$, constructs compact subsets of $[0,1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $\{q^n:n\ge1\}$, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.
Other papers in this family:
- The dyadic case of the Erdős similarity conjecture · September 25, 2026