Problem 120

Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$ ?

Source: erdosproblems.com/120

Combinatorics

Wiki page

Status

Open

No claim settles this problem.

Open: the site labels the problem OPEN (page last edited 23 January 2026), and its commentary names the dyadic sequence $\{1,1/2,1/4,\ldots\}$ as an open special case. That case is settled by an accepted partial claim of the OpenAI mathematics release, the dyadic case, whose Lean declaration this corpus's verification built and audited. Three pending partial claims claim further special cases: the release's geometric-progression case for each fixed ratio, of which only the ratio $1/2$ is formally verified; and two arXiv preprints of 2026 by Iosevich and coauthors, on sums and differences of a geometric sequence and an infinite set and on sets supporting a Rajchman measure. None touches the general question, and the one claim of the full conjecture, by Cruz, Lai and Pramanik in 2020, was withdrawn by its authors, so the derived standing is open.

JLM24: Y. Jung and C.-K. Lai and Y. Mooroogen, Fifty years of the Erdős similarity conjecture. arXiv:2412.11062 (2024). Library home: Jung, Lai and Mooroogen 2024.
OAI26a: OpenAI, The dyadic case of the Erdős similarity conjecture. OpenAI Math Release preprint, 25 September 2026; a release manuscript with no journal or arXiv record. Theorem 1.1, p. 2. Library home: OpenAI 2026.
OAI26b: OpenAI, The geometric case of the Erdős similarity conjecture. OpenAI Math Release preprint, 5 October 2026; a release manuscript with no journal or arXiv record. Theorem 1.1, p. 1. Library home: OpenAI 2026.
St20: Steinhaus, Hugo, Sur les distances des points dans les ensembles de mesure positive. Fund. Math. 1 (1920), 93-104. DOI: 10.4064/fm-1-1-93-104. Library home: Steinhaus 1920.
Er78: Erdős, P., Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space. Real Anal. Exchange 4 (1978/79), no. 2, 113--138. Library home: Erdős 1978.
Sv00: Svetic, R. E., The Erdős similarity problem: a survey. Real Anal. Exchange (2000/01), 525-539.

Problem 120

Status

Proof claims

Discussion