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. (Lean)

Manuscripts (2)

The geometric case of the Erdős similarity conjecture

October 5, 2026 · source · cite as \cite{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026}

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.

The dyadic case of the Erdős similarity conjecture

September 25, 2026 · source · cite as \cite{OAI:The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026}

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\ge1\}. The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.

Result 084 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.