An affine copy of the dyadic sequence in every set of positive measure
The Erdős similarity problem asks whether for every infinite set $X \subset \mathbb{R}$ there is a set of positive measure containing no affine copy $\{a + bx : x \in X\}$, $b \ne 0$, of $X$; the question below is the case of the sequence $1, \frac12, \frac14, \dots$.
Problem. Let $A \subset \mathbb{R}$ have positive measure. Must $A$ contain an affine copy of $\{1, \frac12, \frac14, \dots\}$?
Source: A special case of the Erdős similarity problem (P. Erdős, 1974); Erdős problem #120. Listed as Problem 94 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Keywords
- affine copies, Erdős problems, Erdős similarity problem, Lebesgue measure
References
@article{svetic2000survey,
author = {Svetic, R. E.},
title = {The {E}rd{\H o}s similarity problem: a survey},
journal = {Real Analysis Exchange},
volume = {26},
number = {2},
pages = {525--539},
year = {2000},
doi = {10.2307/44154057}
}
@article{jung2025fifty,
author = {Jung, Yeonwook and Lai, Chun-Kit and Mooroogen, Yuveshen},
title = {Fifty years of the {E}rd{\H o}s similarity conjecture},
journal = {Research in the Mathematical Sciences},
volume = {12},
number = {1},
pages = {Paper No. 9},
year = {2025},
doi = {10.1007/s40687-025-00495-w}
}
@misc{openai2026dyadic,
author = {{OpenAI}},
title = {The dyadic case of the {E}rd{\H o}s similarity conjecture},
howpublished = {\url{https://github.com/openai/math/tree/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026}},
year = {2026},
note = {AI-generated manuscript, 25 September 2026, with a Lean proof}
}
@misc{jiang2026null,
author = {Jiang, Kan and Xi, Lifeng},
title = {The {E}rd{\H o}s similarity problem for null sequences with adjacent ratios bounded below},
year = {2026},
eprint = {2610.11449},
archivePrefix = {arXiv}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 94; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
Let $(a_n)$ be a positive null sequence with $\liminf_{n\to\infty}a_{n+1}/a_n>0$. For every $η>0$, we construct a compact set $E\subset[0,1]$ with Lebesgue measure greater than $1-η$ containing no nontrivial affine copy of $\{a_n:n\ge1\}$. No monotonicity is assumed. The proof adapts a finite-tree construction for geometric sequences to grids chosen from a subsequence with two-sided ratio bounds. We also obtain one such set …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.11449v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01046, "An affine copy of the dyadic sequence in every set of positive measure", clopen.solutions.