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Claim. Write $A_S(x)$ for the Lebesgue measure of $S\cap(0,x)$. Theorem 1.1 of Boon Suan Ho, Erdős's ratio-growth problem and a quantitative Haight-type theorem (manuscript dated April 2026, posted at boonsuan.github.io on 19 April 2026 and revised there on 3 May 2026; both versions are linked above at their commits, and this page describes the revision, whose abstract and Theorem 1.1 agree with the first posting), states: for a non-decreasing $F:[1,\infty)\to[0,\infty)$ with $F(x)\to\infty$, there is a measurable $S\subset(0,\infty)$ of infinite measure in which $x/y$ is never an integer for distinct $x,y\in S$ and $A_S(x)\ge F(x)$ for all sufficiently large $x$ exactly when

$$\int_1^\infty\frac{F(x)}{x^2}\,dx<\infty.$$

This answers the question of [[problems/analysis/E1195/_index|Problem 1195]]: growth like $x/(\log x)^{1+\varepsilon}$ is attainable, after changing the target on a bounded interval, and growth like $x/\log x$ is not, which sharpens Erdős's remark that $A_S(x)=o(x)$. The manuscript notes that Erdős's strict inequality $A_S(x)>F(x)$ follows by applying the theorem to $F+1$. The necessity comes from the ratios $2,4,8,\ldots$ alone: folding the dyadic annuli of $S$ into one annulus bounds $\int_S dt/t$ by $\log2$, and Tonelli's theorem turns that into the convergence of $\int_0^\infty A_S(x)x^{-2}\,dx$. The sufficiency is the main work. The manuscript describes its method as working in logarithmic coordinates, where a forbidden ratio becomes a translation, placing a rapidly oscillating comb in each new annulus, using a simultaneous recurrence so that the finitely many translations relevant at that scale nearly preserve the comb, and deleting a summable proportion of the earlier blocks to remove every future conflict. Theorem 1.2 gives the same criterion for any locally finite set of forbidden ratios in $(1,\infty)$ containing a full geometric progression, Theorem 1.3 proves the sufficient direction for every nonempty locally finite ratio set, a quantitative form of Haight's 1975 theorem, and Theorem 1.4 treats growth along a sequence of scales.

AI system. The manuscript's disclosure states that GPT-5.4 Pro was used substantially and contributed materially to generating and shaping the mathematics, through an extended iterative dialogue, while the author set the aims, structure and revisions and takes responsibility for the text. The disclosure appears in the revision of 3 May 2026; the posting of 19 April 2026 carries none, and the forum announcement of that day says that the proof was found with the assistance of GPT-5.4 Pro and that its summary was written with the system's help and vetted by the author. The site credits the result to Suan and to GPT. The author signs as Boon Suan Ho, and this page is filed under the surname Ho.

Discussion. In the thread, on 28 April 2026, Terence Tao gave a shorter construction for the sufficient direction of the criterion through Bohr sets and simultaneous Dirichlet approximation, and observed that the manuscript's Lemma 3.2 is essentially the simultaneous Dirichlet approximation theorem and that its Bohr sets are used without being named. The revision of 3 May 2026 thanks Tao for the comments and says that it has not yet been revised in response to the thread. Other thread comments report checks made with AI systems; they are not review.

Acceptance. The site's curator, Thomas F. Bloom, marks Problem 1195 solved and credits this characterization, the reviewed evidence. The manuscript is not refereed. A Lean file in Boris Alexeev's lean-proofs repository, linked above at a pinned commit, declares itself a formalization of this result, naming Boon Suan Ho and GPT-5.4 Pro as its informal authors and Codex and GPT-5.6 Sol as its formal authors; its theorem erdos_1195 states the criterion for a function that is nonnegative, nondecreasing and tends to infinity on $[1,\infty)$, and at that commit the file contains no sorry. This corpus has not built or audited it, so no formalized evidence is listed. No statement file for the problem exists in formal-conjectures (none on main on 2026-10-07). The page is dated by the forum announcement, the manuscript's first posting.

Depends on. Nothing beyond the cited manuscript.