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Statement

Proposition 1.7 (Upper bounds), pp. 6--7, states:

For any $n\in\mathbb N$, one has

$$> f_{\mathrm I}(n)\ll n^{3/5+O(1/\log\log n)} >$$

and

$$> f_{\mathrm{II}}(n)\ll n^{2/5+O(1/\log\log n)}. >$$

In particular, from this and (1.4) one can conclude that for any prime $p$ one has

$$> f(p)\ll p^{3/5+O(1/\log\log p)}. >$$

Source. Elsholtz and Tao, arXiv:1107.1010v6, pp. 6--7 (the statement begins at the foot of p. 6 and ends on p. 7); read on the page image of p. 7 and in the text layer of p. 6. Proved in Section 3 (per p. 7). Published as J. Aust. Math. Soc. 94 (2013), 50--105, DOI 10.1017/S1446788712000468; the published version was not compared.

Read depth. Claims checked: the statement and its consequence for primes were read clause by clause; the proof (Section 3) was not read. The paper compares the bound with Browning and Elsholtz's $f(n)\ll_\varepsilon n^{2/3+\varepsilon}$ for all $n$ (its [8]) and says that Proposition 1.7 "appears to be the limit of what one can obtain purely from the divisor bound (A.6) alone" (p. 7).

Dependencies

The parametrizations of Type I and Type II solutions in Section 2 (Propositions 2.2 and 2.6, with the size bounds of Lemma 2.8) and the divisor bound (the paper's (A.6)).

Bears on

• Problem 242: the site's "$f(p)\le p^{3/5+o(1)}$ for all primes $p$"; Elsholtz and Planitzer's Corollary 1 extends the bound $O_\varepsilon(n^{3/5+\varepsilon})$ to all denominators $n$.