Erdős Problem #953 — Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert ot\in \mathbb{Z}$ for any distinct $a,b\in A$.

Erdős Problem #953 asks how large a measurable subset of a disk in the plane can be if it contains no pair at an integer distance. The problem remains unresolved.

Known results

• For disk radius $X$, the maximum measure satisfies the elementary upper bound $f(X)=O(X)$.
• Thickening Sárközy’s construction gives the lower bound $f(X)=\Omega(X^{1/2-o(1)})$.
• For an open-set variant with a fixed number of connected components, an exact extremal formula is known, but it does not settle the measurable-set problem.

May 2026 bounds

The paper “Point sets avoiding near-integer distances” formulates the measurable quantity $f(X)$ and records the bounds above, while explicitly leaving sharper measurable-set bounds open. Its acknowledgment of Gemini 3.1 Pro and ChatGPT5 concerns generating proof ideas, not solving this problem.

Current status (as of August 2026): Partial progress consists of $f(X)=O(X)$ and $f(X)=\Omega(X^{1/2-o(1)})$; the correct growth rate and extremal measure remain open.