The Erdős similarity conjecture for geometric progressions
For every q in (0, 1), a compact subset of [0, 1] of measure as close to 1 as we like contains no affine copy x + s{q^n : n ≥ 1}, s ≠ 0, so geometric progressions are not measure universal. Random tables on nested dyadic grids route every point through a finite ordered tree; the first default choice on the route of a center offers many independent chances to hit a copy, windows of indices with gaps keep earlier decisions unchanged, and finitely many dilations represent all of them. The course is an exposition of OpenAI's proof (2026) of the geometric-progression case of the Erdős similarity conjecture.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Measure-universal sets and periodic hitting sets
- Dyadic grids and windows on a tree
- Random routing and independent tests
- Every scale and every center
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Lebesgue measure on the line: countable additivity, translation invariance and outer regularity (core course Measure and Integration).
- Finite probability spaces: independence and conditioning.