Erdős Problem #953

Erdős problem #953 · erdosproblems.com/953

What is the largest possible measure of a subset of a radius-$R$ disk in $\mathbb{R}^2$ containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives $M(R) \ll R^{1/2}$; with Sárközy's lower construction, $M(R) = R^{1/2 + o(1)}$.

Result

Proved

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Geometric Measure Theory

Posed by

—

Year posed

1977

Years open

49y

Solved

2026-04-27

Model

GPT-5.5 Pro

Vendor

OpenAI

Collaborators

—

Verification

Unreviewed

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article