Result 084, Real and complex analysis

The geometric case of the Erdős similarity conjecture

For every fixed q∈(0,1)q\in(0,1), constructs compact subsets of [0,1][0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of {qn:n≥1}\{q^n:n\ge1\}, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

Lean formalization Proof

The bigger picture

Why it matters

A set can occupy almost all of an interval yet avoid every shifted and rescaled version of a particular infinite geometric progression. The manuscript reports this for every fixed shrinking ratio, including reflected copies.

What changes?

For each fixed ratio q strictly between zero and one and each tolerance eta strictly between zero and one, the manuscript constructs a compact (closed and bounded) subset of [0,1] with length greater than 1 minus eta. It contains no copy of the infinite sequence q, q squared, q cubed, and so on under any translation and any nonzero real scaling, positive or negative. The set may depend on q; the claim is not simultaneous across ratios.

What does that help mathematicians do?

This rules out any guarantee based only on a set's length, short of full length, that it contains a copy of a prescribed infinite geometric progression. The obstruction persists even when the set is closed and almost fills the interval. Unlike the dyadic result, which concerns ratio one-half, this covers each fixed ratio, but not arbitrary infinite sequences.

Are there practical applications?

Its immediate value is foundational: it clarifies the limits of using measure, the mathematical notion of length, to force exact infinite patterns. For researchers studying the Erdős similarity conjecture, it settles the geometric family while leaving the broader question for other infinite sequences unresolved. The supplied sources describe no practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The geometric case of the Erdős similarity conjecture

October 5, 2026 13 pages

We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed q∈(0,1)q\in(0,1) and every η∈(0,1)\eta\in(0,1), we construct a compact set Eq,η⊆[0,1]E_{q,\eta}\subseteq[0,1] of measure greater than 1−η1-\eta containing no nontrivial affine copy of {qn:n≥1}\{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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026,
  author = {{OpenAI}},
  title = {{The geometric case of the Erd\H{o}s similarity conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf}{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026}},
  year = {2026}
}

The dyadic case of the Erdős similarity conjecture

September 25, 2026 16 pages

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}\{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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026,
  author = {{OpenAI}},
  title = {{The dyadic case of the Erd\H{o}s similarity conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf}{OAI:The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/084.md.

The geometric case of the Erdős similarity conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the dyadic case of the Erdős similarity conjecture. For every 0<η<10<\eta<1, it constructs a compact set E⊂[0,1]E\subset[0,1] of measure greater than 1−η1-\eta that contains no affine copy of {2−n:n≥1}\{2^{-n}:n\ge1\}. Explicitly, for every translation xx and every nonzero real dilation ss, some point x+s2−nx+s2^{-n} lies outside EE. Both signs of ss are included.

Comparator links

Result Comparator statement
Positive-measure avoidance of every affine dyadic sequence DyadicAvoidance.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.