---

Statement

Write $\|t\|$ for the distance from a real number $t$ to the nearest integer and $\phi$ for Euler's totient function. Theorem 1.1 (the manuscript's "Weak inhomogeneous Duffin–Schaeffer"). Let $\gamma$ be a fixed real number and $\psi:\mathbb N\to[0,\infty)$ a function taking finite values, and suppose that

$$\sum_{q=1}^{\infty}\frac{\phi(q)}{q}\,\psi(q)=\infty.$$

Then the real numbers $x$ for which $\|qx-\gamma\|<\psi(q)$ has only finitely many solutions in positive integers $q$ form a Lebesgue-null set.

The exceptional null set may depend on $\gamma$ and on $\psi$. Writing $|qx-a-\gamma|<\psi(q)$ with $a$ the integer nearest to $qx-\gamma$, the theorem asks nothing of $a$: it need not be coprime to $q$, which is what "weak" means here. The manuscript places no monotonicity condition on $\psi$ and no Diophantine condition on $\gamma$, and states that the conclusion is a divergence statement only, with no convergence converse and no quantitative asymptotic. It identifies the statement with Question 1.2 of Yu (2021), Conjecture 1.22 of Chow and Technau (2024) and Conjecture 2 of Beresnevich, Hauke and Velani (2024), and it records that the coprime-numerator version is false in general, citing two preprints of 25 September 2026: Hauke-Treuer, Maynard and Pollington (Theorem 1) for counterexamples at all nonzero rational shifts and at some Liouville shifts, and He and Liao (Theorems 1--2) for all nonzero rational shifts and a residual set of real shifts.

Source. OpenAI, The Weak Inhomogeneous Duffin–Schaeffer Conjecture, release folder preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026; TeX source sections/introduction.tex, label thm:main, lines 9--21; PDF p. 3 of 87. The proof occupies Sections 2--10 (PDF pp. 6--85) and closes in sections/joins.tex lines 787--853 (PDF pp. 84--85, ending on p. 85). Read on 2026-10-07. The card records the provenance and the release's own statements about how the manuscript was produced.

Read depth. Claims checked: the statement, its hypotheses and the surrounding paragraph on the meaning of "weak" were read clause by clause in the TeX source. The proof was read for its structure only, as summarized below, and no step was checked. Nothing here is independently reviewed; the manuscript carries no Lean formalization in the release, no arXiv version and no refereed publication known to this corpus.

Proof pointer

The argument is a contradiction argument on finite blocks of denominators. Section 2 first disposes of the case where $\psi(q)$ does not tend to zero (Lemma 2.1) and then, assuming a positive-measure set $A$ avoids all approximation intervals from some index on, cuts the tail into disjoint finite blocks whose totient-weighted mass lies between $w$ and $2w$ for a small fixed $w$ (Lemma 2.2). Everything after that is an estimate on one such block, uniform in the block.

Sections 3 and 4 select the rows (denominators) to use and fix their initial weights. Each row is assigned a center, an integer multiple of its denominator chosen by a cost minimization (Definition 3.1, Lemma 3.5); a density bound above hosts (Lemma 3.2) and a tensor kernel inequality (Lemma 2.6, derived from a one-coordinate concentration lemma, Lemma 2.5, which the manuscript attributes in form to Green–Walker and Hauke-Treuer, Vazquez and Walker) make pair sums over centers summable (Lemma 3.3, Lemma 3.7). Rows whose centers are themselves ("raw" rows) are handled directly by a second-moment argument (Lemma 3.4). A hierarchy of about thirty fixed parameters is ordered in Section 3.5. Section 4 removes a negligible set of deficit masks (Lemma 4.2) and prescribes "hard" numerator exclusions at tag primes and at primes of a recognized rational model of the projected phase (Construction 4.3, Definition 4.4), with a uniform lower bound on the retained mean (Lemma 4.6) and an arbitrarily small relative cost for additional "soft" gcd tests (Lemma 4.7).

Section 5 defines the adaptive construction: each row first appears on a finer projected grid, and as the primes of its center are read in increasing order the grid is restricted and weights are updated by a fractional matching rule that preserves first mass (Lemma 5.1). The section then isolates three quantitative inputs (Propositions 5.2, 5.3 and 5.4: first-sharing comparisons, sparse-step alignments, and the existence of chronological partitions), proves a variance budget (Lemma 5.5) and a uniform second-moment bound for the resulting weighted sum assuming those inputs (Proposition 5.6), constructs dominating "virtual" product weights with exact product means (Proposition 5.7, Corollary 5.8) and their equidistribution on fixed intervals (Lemma 5.9), and concludes (Proposition 5.10) that the inputs imply Theorem 1.1: approximate $A$ by a finite union of intervals, use equidistribution for a lower bound on the integral over that union and the second moment for the approximation error, and contradict the fact that the sum vanishes on $A$.

Sections 6--10 prove the three inputs. Section 6 supplies the arithmetic: a Selberg upper-bound sieve for close grid pairs (Lemma 6.1, proved in full), a rotation alternative that converts frequent close returns of $n\alpha$ into a rational approximation of $\alpha$ (Lemma 6.2), a weighted least-common-multiple bound in the common-pivot style of Green–Walker and Hauke-Treuer–Vazquez–Walker (Proposition 6.3) and a small-scale extraction proposition (Proposition 6.4, with sampling Corollary 6.5) that either gives a summable saving or produces a structured family of denominator pairs. Section 7 proves the simultaneous-birth part of Proposition 5.2 (Proposition 7.1): exact coincidences after replacing phases by rational models are charged linearly to first mass (Lemma 7.4), and the structured families from Section 6 are excluded case by case (Lemmas 7.9--7.11) using tag exclusions, a static "hole" discard near low rational grids (Construction 7.7, Lemma 7.8) and model recognition. Section 8 proves Proposition 5.3 (Proposition 8.1) and builds deterministic interval "trains" covering every class-joining segment (Construction 8.2). Section 9 proves Proposition 5.4 by placing new cell boundaries outside inherited class hulls using those trains (Lemmas 9.1--9.4); boundary candidates are drawn uniformly at random, and the loss bound is in expectation over these draws. Section 10 proves the remaining first-join comparisons (Proposition 10.1) by a backward expansion of actual weights (Lemma 10.3) and a capacity argument charging covariance terms to successful plus entries (Lemma 10.6), then assembles all inputs in chronological order and records the parameter margins (proof of Theorem 1.1, pp. 84--85).

Dependencies

External results cited at statement level: the mass transference principle of Beresnevich and Velani (2006) is used only for Corollary 1.2, not for Theorem 1.1. The Selberg sieve (Selberg 1947, Uchiyama 1962) and the one-coordinate concentration lemma (Green and Walker 2021, Lemma 2.1; Hauke-Treuer, Vazquez and Walker 2026, Lemma 3.2) are cited for their form but reproved in the text; the common-pivot structure of Proposition 6.3 is attributed to Koukoulopoulos–Maynard (2020), Green–Walker (2021, Proposition 1.2 and Section 3) and Hauke-Treuer, Vazquez and Walker (2026, Proposition 2.1), and the manuscript states that its weighted version is proved locally. The Pollington–Vaughan overlap estimate (1990), as restated in Koukoulopoulos–Maynard Lemma 5.3, is named as the homogeneous antecedent of Lemma 7.6 and explicitly not used as an input. Elementary prime-counting bounds (Lemma 2.4), the Chinese remainder theorem, Cauchy–Schwarz, Hölder and Markov inequalities are used throughout. None was checked here.

Bears on

• Problem 999: at $\gamma=0$ the theorem is the unrestricted-numerator divergence half of the problem's statement and follows from the problem's proved coprime-numerator form (Koukoulopoulos–Maynard); for $\gamma\ne0$ it is a claimed extension of that weak form to every fixed real shift, a variant the page does not ask about. The claim is unverified here, and the page's status rests on the acceptance evidence it records, not on this manuscript.