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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; Corollary 3 is stated on p. 4.

Statement

Let $\psi:\mathbb N\to[0,1/2]$. Let $\mathcal A$ be the set of $\alpha\in[0,1]$ for which

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

has infinitely many solutions in coprime integers $a$ and $q$, and let

$$s=\inf\Bigl\{\beta\in\mathbb R_{\ge0}:\ \sum_{q=1}^{\infty}\varphi(q)\bigl(\psi(q)/q\bigr)^{\beta}<\infty\Bigr\}.$$

Then the Hausdorff dimension of $\mathcal A$ is $\dim_{\mathcal H}(\mathcal A)=\min(s,1)$.

Unlike Theorems 1 and 2, the corollary restricts $\psi$ to values in $[0,1/2]$.

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on the pages of the edition named above. The corollary has no proof in the paper beyond its attribution, and the cited result of Beresnevich and Velani was not read. Nothing here is independently reviewed.

Proof pointer

No proof is written out. The paper says (p. 4) that Beresnevich and Velani proved that the Duffin-Schaeffer conjecture implies a Hausdorff measure version of itself, and that the corollary is immediate from their results combined with Theorem 1.

Dependencies

• Theorem 1, with the Hausdorff measure transference of Beresnevich and Velani, Ann. of Math. (2) 164 (2006), 971--992, which is not in the corpus.

Bears on

• Problem 999: context only. The problem concerns the Lebesgue measure of the set of $\alpha$ with infinitely many reduced approximations; the corollary gives that set's Hausdorff dimension, $\min(s,1)$, for $\psi$ with values in $[0,1/2]$.