The geometric case of the Erdős Similarity Conjecture. A human exposition of the OpenAI proof

Abstract

For a fixed $q\in(0,1)$, can a set of positive Lebesgue measure avoid every translated and dilated copy of $\{q^n:n\ge1\}$? 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.