Erdős Problem #1195

The problem

Let $S\subset \mathbb{R}$ be a set of infinite measure such that $x/y$ is never an integer for all distinct $x,y\in S$.

How fast can $\lvert S\cap (0,x)\rvert$ tend to infinity?

Statement quoted from erdosproblems.com/1195, maintained by Thomas Bloom. Original source: [Er80,p.101]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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