Problem 1195

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claims/: The 1 claim page of Problem 1195, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Status. Solved, the site's label. Boon Suan Ho, working with GPT-5.4 Pro, characterized the attainable growth: for non-decreasing $F\to\infty$, some admissible $S$ has measure at least $F(x)$ in $(0,x)$ for all large $x$ exactly when $\int_1^\infty F(x)x^{-2}\,dx<\infty$. The accepted claim is Ho's sharp growth criterion, credited by the site's curator and not refereed; a Lean formalization in Boris Alexeev's repository is linked on the claim page, and this corpus has not built or audited it.

Source. erdosproblems.com/1195, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1195, https://www.erdosproblems.com/1195.

References.

• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
• [Ha70] Haight, J. A., A linear set of infinite measure with no two points having integral ratio. Mathematika (1970), 133-138.
• [Sc69] Schmidt, W. M., Disproof of some conjectures on Diophantine approximations. Studia Sci. Math. Hungar. (1969), 137-144.
• [Sz71] Szemerédi, E., On a problem of W. Schmidt. Studia Sci. Math. Hungar. (1971), 287-288.

Formalization. No statement file for the problem exists in formal-conjectures (none on main on 2026-10-07; the site's page lists no formalised statement and the community database records the problem unformalized). A Lean formalization of Ho's theorem in Boris Alexeev's lean-proofs repository, naming Boon Suan Ho and GPT-5.4 Pro as its informal authors and Codex and GPT-5.6 Sol as its formal authors, is linked on the claim page; this corpus has not built or audited it, and the standing rests on the manuscript and the curator's credit.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• schmidt_1969_disproof_conjectures_diophantine_approximations
• schmidt_1969_disproof_conjectures_diophantine_approximations / question_p138