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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 2 is stated on p. 3.

Statement

Let $\psi:\mathbb N\to\mathbb R_{\ge0}$, and let $\mathcal K$ be the set of $\alpha\in[0,1]$ for which

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

has infinitely many solutions $(a,q)\in\mathbb Z^2$ with $0\le a\le q$. Define $\psi^*:\mathbb N\to\mathbb R_{\ge0}$ by

$$\psi^*(q)=\varphi(q)\,\sup\{\psi(n)/n:\ n\in\mathbb N,\ q\mid n\}.$$

Then:

(a) if $\sum_{q=1}^{\infty}\psi^*(q)<\infty$, then $\lambda(\mathcal K)=0$;

(b) if $\sum_{q=1}^{\infty}\psi^*(q)=\infty$, then $\lambda(\mathcal K)=1$.

Here $\lambda$ is Lebesgue measure and the fractions $a/q$ need not be reduced. The paper presents the theorem as Catlin's conjecture, obtained as a direct corollary of Theorem 1, and as an extension of Khinchin's theorem, which assumes $q\psi(q)$ decreasing (pp. 1 and 3).

Read depth. Claims checked: the statement was read clause by clause and the deduction in Section 2 was followed on the pages of the edition named above. Nothing here is independently reviewed.

Proof pointer

Section 2 (pp. 5--6), following Catlin. If $\psi(q)\ge1/2$ for infinitely many $q$, then $\mathcal K=[0,1]$ and the series of $\psi^*$ diverges, which the paper checks along a sparse subsequence. Otherwise $\psi$ may be capped at $1/2$, the supremum becomes a maximum, and with $\xi(q)/q=\max_{q\mid n}\psi(n)/n$ the set $\mathcal C$ of $\alpha$ with infinitely many reduced approximations within $\xi(q)/q$ agrees with $\mathcal K$ off the rationals: a reduced approximation for $\xi$ scales up to one for $\psi$, and an approximation for $\psi$ reduces to one for $\xi$. Since $\psi^*(q)=\varphi(q)\xi(q)/q$, part (b) is Theorem 1 applied to $\xi$ and part (a) is the convergence implication (1.5) (p. 2).

Dependencies

• Theorem 1 gives part (b).

Bears on

• Problem 999: context only. The problem asks about reduced fractions $p/q$ with $(p,q)=1$; Theorem 2 is the companion statement for fractions that need not be reduced, with $\psi^*$ in place of $\psi\varphi(q)/q$.