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Claim. V. Y. Wang, Sums of cubes and the Ratios Conjectures, arXiv:2108.03398v2 (19 April 2023), Theorem 1.3 (source card wang_2021_sums_cubes_ratios_conjectures): assume Conjectures 1.2, 1.4 and 1.5; then $N_F(X)\ll X^3$ for $F=x_1^3+\cdots+x_6^3$, and, with $F_0=x^3+y^3+z^3$, "Let $S\subseteq\mathbb{Z}_{\geq 0}$. If $S$ has positive lower density in $\mathbb{Z}_{\geq 0}$, then so does $F_0(S^3)$." With $S=\mathbb{Z}_{\geq 0}$, the integers $x^3+y^3+z^3$ with $x,y,z\ge0$ have positive lower density. Each of them other than $0$ is a sum of at most three positive cubes, and every positive cube is $3$-powerful, so under the three conjectures the integers that are sums of at most three $3$-powerful numbers have positive lower density and do not have density $0$. That answers the second question of Problem 940 no at $r=3$, conditionally. The first version of the preprint, posted on 7 August 2021 as Approaching cubic Diophantine statistics via mean-value $L$-function conjectures of Random Matrix Theory type, already states in its abstract that, under its hypotheses (1)--(4), a positive fraction of integers lie in $\{x^3+y^3+z^3:x,y,z\in\mathbb{Z}_{\geq 0}\}$.

Hypotheses. As the second version numbers them: Conjecture 1.2 (HW2), automorphy, and no zeros in $\operatorname{Re}(s)>1/2$, for the Hasse--Weil $L$-functions of its list (1.7); Conjecture 1.4 (R2$'$), a log-free second-moment bound of Ratios type, display (1.10); and Conjecture 1.5, a square-free sieve conjecture for the discriminant polynomial. All three are unproved, so the result is conditional and settles no case of the problem.

Scope. The result concerns $r=3$ and the density question only; it says nothing about the first question or about any $r\ge4$. The formal-conjectures statement erdos_940 asks whether the representable set has density $0$ for every $r\ge3$ at once, and its variant erdos_940.variants.three_cubes asks it for sums of at most three nonnegative cubes; under the three conjectures both are answered no.

Acceptance. None. The result is an arXiv preprint with no journal version known, its hypotheses are unproved, and the site does not credit it: its commentary (page last edited 3 November 2025) says that at $r=3$ it is not even known whether the sums of at most three cubes have density $0$.

Depends on. No page of this wiki; the claim rests on the preprint above.