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Source. Theorem 1.3, p. 3, of Victor Y. Wang, Sums of cubes and the Ratios Conjectures, arXiv:2108.03398v2 (19 April 2023), the edition named on the source card. The hypotheses are Conjecture 1.2 (p. 3), Conjecture 1.4 (p. 4) and Conjecture 1.5 (p. 4).

Read depth. Claims checked: the statement, the three conjectures it assumes and the notation of §1 were read clause by clause on pp. 2--4; the proof in §10.2 (p. 55) was read for its structure. Nothing here is independently reviewed.

Statement

Write $F_0(x,y,z)=x^3+y^3+z^3$ and, for a cubic form $F$ in $m$ variables,

$$N_F(X)=\#\{\mathbf x\in[-X,X]^m: F(\mathbf x)=0\}$$

(display (1.2), p. 2), counting integer points. For a nonzero integer vector $\mathbf c$, $V_{\mathbf c}$ is the variety $F(\mathbf x)=\mathbf c\cdot\mathbf x=0$ in $\mathbb P^{m-1}$, $\Delta(\mathbf c)$ is the discriminant polynomial of §2 (display (2.1), p. 8), and $\mathcal S_1$ is the set of $\mathbf c\in\mathbb Z^m$ with $\Delta(\mathbf c)\ne0$ (display (1.6), p. 3).

Theorem 1.3 (p. 3). Let $F=x_1^3+\cdots+x_6^3$, and assume Conjectures 1.2, 1.4 and 1.5. Then

$$N_F(X)\ll X^3$$

(display (1.8)). Moreover, "Let $S\subseteq\mathbb Z_{\ge0}$. If $S$ has positive lower density in $\mathbb Z_{\ge0}$, then so does $F_0(S^3)$." (p. 3).

The three hypotheses, as the paper states them:

• Conjecture 1.2 (HW2) (p. 3). For each $\mathbf c\in\mathcal S_1$ and each of the Hasse--Weil $L$-functions $L(s,V_{\mathbf c})$, $L(s,V_{\mathbf c},\otimes^2)$, $L(s,V_{\mathbf c},\mathrm{Sym}^2)$, $L(s,V_{\mathbf c},\wedge^2)$, $\zeta(s)$ and $L(s,V)$ of list (1.7), where $V$ is the hypersurface $F=0$: there are an integer $d\ge1$ and an isobaric automorphic representation $\Pi$ of $\mathrm{GL}_d(\mathbb A_{\mathbb Q})$ whose local factors agree with those of the $L$-function at every place, gamma factor included, and $L(s,\Pi)$ has no zeros in $\operatorname{Re}(s)>1/2$.
• Conjecture 1.4 (R2$'$) (p. 4). For even $m$, with $\Phi^{\mathbf c,1}(s)=1/\zeta(2s)L(s+1/2,V)L(s,V_{\mathbf c})$ (display (1.9)): for every entire $f$ with $f(s)\ll_{f,b}(1+|\operatorname{Im}(s)|)^{-b}$ on the strip $0\le\operatorname{Re}(s)\le2$ for all $b\in\mathbb Z_{\ge1}$, all reals $Z,N\ge1$ with $N\le Z^3$, and every $\sigma_0\in(1,2)$,

$$\sum_{\mathbf c\in\mathcal S_1\cap[-Z,Z]^m}\Bigl|\int_{(\sigma_0)}ds\,\Phi^{\mathbf c,1}(s)f(s)N^s\Bigr|^2 \ll_F Z^mN\sup_{0\le\sigma\le2}\int_{\mathbb R}dt\,(1+|t|)^2|f(\sigma+it)|^2$$

(display (1.10)), the contour running from $\sigma_0-i\infty$ to $\sigma_0+i\infty$.

• Conjecture 1.5 (SFSC$_{p,3}$) (p. 4). There is a real $\eta_0=\eta_0(\Delta)>0$ such that for all reals $Z,P\ge1$ with $P\le Z^{3/2}$,

$$\#\{\mathbf c\in\mathbb Z^m\cap[-Z,Z]^m:\ p^2\mid\Delta(\mathbf c)\text{ for some prime }p\in[P,2P]\}\ll_\Delta Z^mP^{-\eta_0}.$$

All three are unproved, so both conclusions are conditional. Unconditionally the paper recalls $N_F(X)\ll_\epsilon X^{7/2}/(\log X)^{5/2-\epsilon}$ for $X\ge2$ when $m=6$ and $F$ is diagonal (p. 3, citing Vaughan), and, assuming automorphy and GRH for $L(s,V_{\mathbf c})$ (Conjecture 1.2 for that function), the Hooley--Heath-Brown bound $N_F(X)\ll_\epsilon X^{3+\epsilon}$ of Theorem 1.1 (p. 3). Theorem 1.3 removes the $\epsilon$.

Proof pointer

§10.2, p. 55. Write $N_F(X)$ as the sixth moment of the cubic Weyl sum over $|x|\le X$ (display (10.20)), split the sum dyadically and apply Hölder ((10.21)--(10.22)), which reduces (1.8) to bounding a smoothed count $N_{F,w}(X)$ with a weight $w$ supported away from the coordinate hyperplanes ((10.23)). The delta-method identity (2.10) splits that count over $\mathbf c\in\mathcal S_0$ and $\mathbf c\in\mathcal S_1$; Theorem 2.5 (p. 10, quoted from Wang's earlier work) bounds the $\mathcal S_0$ part unconditionally, and Theorem 10.5 (p. 52) with $\xi=0$ bounds the $\mathcal S_1$ part, its moment hypotheses Conjectures 9.6 and 9.8 being supplied by Proposition 9.7 and, under Conjecture 1.5, by Proposition 9.9 (p. 46). The density statement follows from (1.8) by a Cauchy--Schwarz argument on the number of representations, which the paper calls standard and does not write out.

Dependencies

Conjectures 1.2, 1.4 and 1.5 as hypotheses; within the paper, Theorem 2.5, Propositions 9.7 and 9.9, and Theorem 10.5.

Bears on

• Problem 940: with $S=\mathbb Z_{\ge0}$, the second conclusion gives the sums $x^3+y^3+z^3$ with $x,y,z\ge0$ positive lower density. Each positive one is a sum of at most three positive cubes, hence of at most three $3$-powerful numbers, so under Conjectures 1.2, 1.4 and 1.5 the sums of at most three $3$-powerful numbers do not have density $0$: a conditional negative answer to the density question at $r=3$ only. The theorem says nothing about the infinitude question or about $r\ge4$, and it settles nothing unconditionally.
• Problem 325: with $S=\mathbb Z_{\ge0}$, the same conclusion gives $f_{3,3}(x)\gg x$ for large $x$, the bound the problem asks for at $k=3$, under the same three unproved conjectures. The theorem says nothing about $k\ge4$.