The Erdős similarity problem

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.

Problem

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

$$aA+b=\{ax+b : x\in A\},\qquad a,b\in\mathbb{R},\ a\neq0\,?$$

Erdős offered $100 for this problem.

Background

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.

What is known

• 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.
• 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.
• 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.
• 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.
• Variants. Jung, Lai and Mooroogen (2024) survey bi-Lipschitz, topological and large-scale versions of the problem.

What would count as progress

• A proof for a new class of fast-decreasing sequences beyond geometric progressions, for example $\{2^{-n^2}\}$ or all sequences with $\limsup a_{n+1}/a_n<1$.
• A counterexample: an infinite set $A$ such that every set of positive measure contains an affine copy of $A$.
• Independent verification and simplification of the recent geometric-sequence arguments.
• A formal proof against the Lean statement in formal-conjectures.

References

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
2. M. N. Kolountzakis, Infinite patterns that can be avoided by measure, Bull. London Math. Soc. 29 (1997), 415–424. doi:10.1112/S0024609397003056
3. R. E. Svetic, The Erdős similarity problem: a survey, Real Anal. Exchange 26 (2000/01), 525–539. doi:10.2307/44154057
4. Y. Jung, C.-K. Lai and Y. Mooroogen, Fifty years of the Erdős similarity conjecture (2024). arXiv:2412.11062
5. OpenAI, The dyadic case of the Erdős similarity conjecture, preprint (2026). PDF; Lean statement
6. OpenAI, The geometric case of the Erdős similarity conjecture, preprint (2026). PDF
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

Source

Imported from Erdős Problems 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.