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Source. Theorem 1.2, p. 2, with the definitions (1.1) on p. 2, of Trevor D. Wooley, Sums of three cubes, II, Acta Arith. 170 (2015), 73--100, read in the arXiv version arXiv:1502.01944v1 named on the source card; labels and pages here are that version's.

Read depth. Claims checked: the statement and the definitions were read clause by clause on p. 2, Lemma 5.1 on p. 16 and the deduction on p. 24 for their structure; the computations of Section 7 were not checked. Nothing here is independently reviewed.

Statement

Let $\mathcal A(P,R)=\{n\in[1,P]\cap\mathbb Z: p\mid n \text{ and } p \text{ prime}\Rightarrow p\le R\}$ be the $R$-smooth numbers of size at most $P$. With $e(z)=e^{2\pi iz}$ put

$$f(\alpha;P,R)=\sum_{x\in\mathcal A(P,R)}e(\alpha x^3), \qquad F(\alpha;P)=\sum_{1\le x\le P}e(\alpha x^3) \qquad(1.1)$$

(p. 2).

Theorem 1.2 (p. 2). Write $\delta_6=0.24871567$. Then there is a positive number $\eta$ such that, whenever $R\le P^\eta$,

$$\int_0^1|F(\alpha;P)^2f(\alpha;P,R)^4|\,d\alpha\ll P^{3+\delta_6}. \qquad(1.2)$$

The paper compares (p. 2) the exponent $\delta_6=0.24941301\ldots$ of the author's earlier work and $\delta_6=\tfrac14+\varepsilon$, any $\varepsilon>0$, from Vaughan's sixth moment for $f(\alpha;P,R)$, and notes that applications need (1.2) with $\delta_6<\tfrac14$. The same $\delta_6$ is the $s=6$ entry of Table 1 (p. 4), and Theorem 1.5 (p. 4) gives $\int_0^1|f(\alpha;P,R)|^6\,d\alpha\ll P^{3+\delta_6}$ under that theorem's hypotheses ($\eta>0$, $P$ sufficiently large in terms of $\eta$, $R\le P^\eta$).

Proof pointer

Section 7, p. 24, with Lemma 5.1 (p. 16). Lemma 5.1 shows that for real $4<t\le8$, if $\delta_6\le\tfrac23$ and $\delta_t\le\tfrac16(t-4)$ are associated exponents (Section 2, p. 5: $U_t(P,R)\ll P^{t/2+\delta_t+\varepsilon}$), then the left side of (1.2) is $\ll P^{3+\delta_6'+\varepsilon}$ with $\delta_6'=2\max\{(8-t+8\delta_t)/(24+t+8\delta_t),\ \delta_6/(4+\delta_6)\}$ (equations (5.1), (5.2)). The computed iteration of Section 7 gives, by convexity, the associated exponent $\delta_t=0.14963020$ at $t=5.392938$, and (5.2) then yields Theorem 1.2.

Dependencies

Lemma 5.1 and the computations of Section 7 of the same paper; Lemma 5.1 uses inequality (5.3) of T. D. Wooley, Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour, Invent. Math. 122 (1995), 421--451 (the paper's reference [24]), and the argument of the author's earlier Sums of three cubes, Mathematika 47 (2000), 53--61 (reference [26]).

Bears on

• Problem 325: only through Theorem 1.1, which the paper deduces from this estimate; that page states the relation.