The geometric case of the Erdős Similarity Conjecture. A human exposition of the OpenAI proof
Abstract
For a fixed q∈(0,1), can a set of positive Lebesgue measure avoid every translated and dilated copy of {qn:n≥1}? OpenAI's release of October 6, 2026 shows that it can, with a compact subset of [0,1] whose measure is arbitrarily close to one. We give a detailed exposition of the proof, using only finite probability and basic measure theory. The proof removes a small open set intersecting every copy, obtained from a random construction: fair random choices, attached to the cells of nested grids, route each point down a finite tree, and at the leaf reached a further random choice removes the point with probability p. The removed set therefore has expected density p, however large the tree, while the branching gives each copy many independent chances to intersect it. Each chance depends on only part of the tree, so its outcome changes at only a controlled number of dilations, and a union bound over finitely many representative dilations then covers every dilation. We also relate the proof to the earlier random-cell arguments going back to Kolountzakis (Bull. London Math. Soc. 29 (1997), 415--424) and to recent work on two-fold sumsets (arXiv:2607.03584) and on Rajchman measures (arXiv:2609.04456), and we show how the method of conditional expectations makes the construction deterministic.
Related
- The Erdős--Sós Conjecture: A visual exposition of the proof discovered by GPT-6 Astra
- The Erdős similarity conjecture and Rajchman measures
- Self-similar swirl between contracting porous walls: the GD1998 exact Navier-Stokes solution revisited in the similarity variables of the OpenAI 2026 forced blow-up construction
- The Deligne--Drinfeld conjecture after OpenAI
- The Erdős similarity problem for null sequences with adjacent ratios bounded below
BibTeX
@misc{iosevich2026geometric,
title = {{The geometric case of the Erdős Similarity Conjecture. A human exposition of the OpenAI proof}},
author = {Alex Iosevich and Alexia Yavicoli},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02454},
eprint = {2611.02454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02454},
}
RIS
TY - UNPB
TI - The geometric case of the Erdős Similarity Conjecture. A human exposition of the OpenAI proof
AU - Alex Iosevich
AU - Alexia Yavicoli
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02454
UR - https://arxiv.org/abs/2611.02454
AB - For a fixed q∈(0,1), can a set of positive Lebesgue measure avoid every translated and dilated copy of {q^n:n≥1}? OpenAI's release of October 6, 2026 shows that it can, with a compact subset of [0,1] whose measure is arbitrarily close to one. We give a detailed exposition of the proof, using only finite probability and basic measure theory. The proof removes a small open set intersecting every copy, obtained from a random construction: fair random choices, attached to the cells of nested grids, route each point down a finite tree, and at the leaf reached a further random choice removes the point with probability p. The removed set therefore has expected density p, however large the tree, while the branching gives each copy many independent chances to intersect it. Each chance depends on only part of the tree, so its outcome changes at only a controlled number of dilations, and a union bound over finitely many representative dilations then covers every dilation. We also relate the proof to the earlier random-cell arguments going back to Kolountzakis (Bull. London Math. Soc. 29 (1997), 415--424) and to recent work on two-fold sumsets (arXiv:2607.03584) and on Rajchman measures (arXiv:2609.04456), and we show how the method of conditional expectations makes the construction deterministic.
ER -