The Erdős similarity problem

Show that for every infinite set A of reals some set of positive measure contains no scaled and shifted copy of A.

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State the problem, from Erdős Problems

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	1	+# The Erdős similarity problem
	2	+
	3	+Every set of real numbers of positive measure contains a scaled and shifted copy of every finite set. Erdős asked whether the same can be true for some infinite set, and conjectured that it never is: for every infinite set $A$ there should be a set of positive measure containing no similar copy of $A$. Posed in 1974, the conjecture was settled for the sequence $1,\tfrac12,\tfrac14,\dots$ only in 2026, and it remains open in general.
	4	+
	5	+## Problem
	6	+
	7	+Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a measurable set $E\subseteq\mathbb{R}$ of positive Lebesgue measure that contains no set of the form
	8	+
	9	+$$
	10	+aA+b=\{ax+b : x\in A\},\qquad a,b\in\mathbb{R},\ a\neq0\,?
	11	+$$
	12	+
	13	+Erdős offered \$100 for this problem.
	14	+
	15	+## Background
	16	+
	17	+Steinhaus showed in 1920 that the answer is negative for finite $A$: a set of positive measure contains an affine copy of every finite set. Erdős conjectured in 1974 that no infinite set has this property. The statement is easy when $A$ is unbounded or dense in some interval, so it suffices to treat a decreasing sequence $a_1>a_2>\dots$ converging to $0$. Falconer (1984) and, independently, Eigen settled sequences that converge to $0$ slowly, in the sense that $a_{n+1}/a_n\to1$; the surveys of Svetic (2000) and of Jung, Lai and Mooroogen (2024) record that no faster-decreasing sequence was known to satisfy the conjecture, with $\{2^{-n}\}$ as the main bottleneck.
	18	+
	19	+## What is known
	20	+
	21	+- Finite sets and slow sequences. Every set of positive measure contains an affine copy of every finite set (Steinhaus, 1920); sequences with $a_{n+1}/a_n\to1$ satisfy the conjecture (Falconer, 1984; Eigen). Kolountzakis (1997) gave a probabilistic criterion that extends this to sequences containing arbitrarily long slowly decaying stretches.
	22	+- Dyadic sequence. The special case $A=\{1,\tfrac12,\tfrac14,\dots\}$ is Problem 94 on Ben Green's list. A September 2026 preprint posted by OpenAI constructs, for every $\eta>0$, a compact set of measure greater than $1-\eta$ containing no affine copy of $\{2^{-n}\}$; its Lean formalisation is linked from the source page.
	23	+- Geometric sequences. An October 2026 preprint from the same source claims the case of every geometric progression $\{q^n\}$, $0<q<1$. It has not been formalised or refereed.
	24	+- Large sets. Iosevich, Kulkarni, Mora Cuéllar, Rojas Aravena and Yavicoli (2026, preprint) proved the conjecture for every set that supports a Rajchman measure, one whose Fourier transform tends to zero. Countable sets support no such measure, so this does not reach sequences.
	25	+- Variants. Jung, Lai and Mooroogen (2024) survey bi-Lipschitz, topological and large-scale versions of the problem.
	26	+
	27	+## What would count as progress
	28	+
	29	+- A proof for a new class of fast-decreasing sequences beyond geometric progressions, for example $\{2^{-n^2}\}$ or all sequences with $\limsup a_{n+1}/a_n<1$.
	30	+- A counterexample: an infinite set $A$ such that every set of positive measure contains an affine copy of $A$.
	31	+- Independent verification and simplification of the recent geometric-sequence arguments.
	32	+- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/120.lean).
	33	+
	34	+## References
	35	+
	36	+1. K. J. Falconer, On a problem of Erdős on sequences and measurable sets, Proc. Amer. Math. Soc. 90 (1984), 77. [doi:10.2307/2044671](https://doi.org/10.2307/2044671)
	37	+2. M. N. Kolountzakis, Infinite patterns that can be avoided by measure, Bull. London Math. Soc. 29 (1997), 415–424. [doi:10.1112/S0024609397003056](https://doi.org/10.1112/S0024609397003056)
	38	+3. R. E. Svetic, The Erdős similarity problem: a survey, Real Anal. Exchange 26 (2000/01), 525–539. [doi:10.2307/44154057](https://doi.org/10.2307/44154057)
	39	+4. Y. Jung, C.-K. Lai and Y. Mooroogen, Fifty years of the Erdős similarity conjecture (2024). [arXiv:2412.11062](https://arxiv.org/abs/2412.11062)
	40	+5. OpenAI, The dyadic case of the Erdős similarity conjecture, preprint (2026). [PDF](https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf); [Lean statement](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DyadicAvoidance.lean)
	41	+6. OpenAI, The geometric case of the Erdős similarity conjecture, preprint (2026). [PDF](https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf)
	42	+7. A. Iosevich, N. Kulkarni, N. Mora Cuéllar, I. Rojas Aravena and A. Yavicoli, The Erdős similarity conjecture and Rajchman measures (2026). [arXiv:2609.04456](https://arxiv.org/abs/2609.04456)
	43	+
	44	+## Source
	45	+
	46	+Imported from [Erdős Problems 120](https://www.erdosproblems.com/120), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.