The geometric case of the Erdős similarity conjecture

What OpenAI claims

For every fixed q ∈ (0,1), constructs compact subsets of [0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of {qⁿ:n ≥ 1}, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio. (Lean)

Papers (2)

1. The geometric case of the Erdős similarity conjecture

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed q ∈ (0,1) and every η ∈ (0,1), we construct a compact set E_q,η ⊆ [0,1] of measure greater than 1-η containing no nontrivial affine copy of {qⁿ:n ≥ 1}, for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.

2. The dyadic case of the Erdős similarity conjecture

   25 September 2026 · Preprint, not peer reviewed · PDF

   We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence {2^-n:n ≥ 1}. The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.