OpenFrontiers
Join
← Back to the index
OP-9273OpenGeometry and topology

Erdős #953 · geometry / distances

Let A⊂{x∈R2:∣x∣<r}A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\} be a measurable set with no integer distances, that is, such that ∣a−b∣∉Z\lvert a-b\rvert \not\in \mathbb{Z} for any distinct a,b∈Aa,b\in A. How large can the measure of …

Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let A⊂{x∈R2:∣x∣<r}A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\} be a measurable set with no integer distances, that is, such that ∣a−b∣∉Z\lvert a-b\rvert \not\in \mathbb{Z} for any distinct a,b∈Aa,b\in A. How large can the measure of AA be?

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

A problem of Erdős and Sárközi. Erdős [Er77c] writes that 'Sárközi has the sharpest results, but nothing has been published yet'.

The trivial upper bound is O(r)O(r). Koizumi and Kovac have observed in the comments that Sárközy's lower bound in [466] can be adapted to give a lower bound of ≫ϵr1/2−ϵ\gg_\epsilon r^{1/2-\epsilon} for all ϵ>0\epsilon>0.

See also [465] for a similar problem (concerning upper bounds) and [466] for a similar problem (concerning lower bounds).

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #953 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.