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Source. Dimitris Koukoulopoulos and James Maynard, On the Duffin-Schaeffer conjecture, Ann. of Math. (2) 192 (2020), no. 1, 251--307, doi:10.4007/annals.2020.192.1.5. Labels and pages here are those of the edition named on the source card, arXiv:1907.04593v3; Theorem 1 is stated on p. 2.

Statement

Let $\psi:\mathbb N\to\mathbb R_{\ge0}$ satisfy

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

Let $\mathcal A$ be the set of $\alpha\in[0,1]$ for which the inequality

$$\left|\alpha-\frac aq\right|\le\frac{\psi(q)}{q}$$

has infinitely many solutions in coprime integers $a$ and $q$. Then $\mathcal A$ has Lebesgue measure $1$.

No monotonicity or regularity of $\psi$ is assumed, and the inequality is non-strict. The proof works with the set $\limsup_{q\to\infty}\mathcal A_q$ of points lying in infinitely many $\mathcal A_q$, where $\mathcal A_q$ is the part of $[0,1]$ covered by the intervals $[(a-\psi(q))/q,(a+\psi(q))/q]$ with $1\le a\le q$ and $\gcd(a,q)=1$ (displays (1.3) and (1.4), p. 2).

The converse half is recorded on p. 2 as (1.5): if the same series converges, then $\lambda(\mathcal A)=0$, by the easy direction of the Borel-Cantelli lemma. Together, (1.5) and Theorem 1 make the measure of $\mathcal A$ equal to $0$ or $1$ according as the series converges or diverges. The paper notes (p. 2) that the implication of Theorem 1 is the conjecture Duffin and Schaeffer posed in 1941, listed as Problem 46 in Montgomery's lectures.

Read depth. Claims checked: the statement, its hypotheses and (1.5) were read clause by clause on the pages of the edition named above. The proof was followed for its structure only; its estimates were not checked. Nothing here is independently reviewed.

Proof pointer

The proof occupies Sections 5 to 14 (pp. 11--45); Section 3 (pp. 6--10) outlines it and Section 4 (pp. 10--11) charts its dependencies.

Section 5 (pp. 11--15). Values $\psi(q)>1/2$ are split off and handled by Lemma 5.2, which the paper takes from Pollington and Vaughan; so one may assume $\psi\le1/2$. Gallagher's zero-one law (Lemma 5.1) then reduces the theorem to $\lambda(\mathcal A)>0$. On a block $[X,Y]$ where the sum of $\psi(q)\varphi(q)/q$ over $X\le q\le Y$ lies in $[1,2]$, a second-moment (Cauchy-Schwarz) argument and the known overlap bound for $\lambda(\mathcal A_q\cap\mathcal A_r)$ (Lemma 5.3, from Pollington and Vaughan) reduce the theorem to a bound on the weighted pairs $(q,r)$ with large greatest common divisor, Proposition 5.4 (p. 12), whose bound is $\ll1/t$ for the pairs in its set $\mathcal E_t$.

Sections 6 to 14. Proposition 5.4 is recast as a statement about weighted bipartite "GCD graphs" (Proposition 6.3, p. 16) with vertex weight $\psi(v)\varphi(v)/v$, and that in turn is reduced to finding a GCD subgraph of high "quality" (Proposition 7.1, p. 19). This subgraph is built by an iteration that passes to subgraphs in which a prime power divides many vertices, a density-increment and compression argument (Propositions 8.1--8.3, pp. 23--24, proved in Sections 12--14 with the lemmas of Sections 9--11). Section 15 (pp. 45--46) explains that the factor $\varphi(v)/v$ in the weights is essential to this approach.

Dependencies

None in the corpus. External inputs named by the paper: Gallagher's zero-one law, and two results of Pollington and Vaughan (the case of large values of $\psi$ and the overlap bound for $\mathcal A_q\cap\mathcal A_r$).

Bears on

• Problem 999: Theorem 1 is the divergence half of the problem's equivalence and (1.5) the convergence half, both for real-valued $\psi\ge0$, for $\alpha\in[0,1]$ and with the non-strict inequality $\le$. The problem states the equivalence for $f:\mathbb N\to\mathbb N$ with the strict inequality $<$; the paper does not address that wording, and the problem's claim page records the translation.