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Source. Theorem 1.9, p. 6, 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 Conjectures 1.2 (p. 3), 1.5 (p. 4), 1.10 and 1.11 (p. 6).

Read depth. Claims checked: the statement and the conjectures it assumes were read clause by clause on pp. 3--6; the proof on pp. 57--58 was read for its structure. Nothing here is independently reviewed.

Statement

Theorem 1.9 (p. 6). Let $m=6$ and let $F$ be diagonal. Assume Conjecture 1.2, Conjecture 1.5 with exponent $\eta_0$, Conjecture 1.10 with exponent $\eta_1$, and Conjecture 1.11 with polynomial $H$. Then there is a real $\delta=\delta(\eta_0,\eta_1,\deg H)>0$ such that

$$E_{F,w}(X)\ll_{F,w}X^{3-\delta}$$

for every $w\in C_c^\infty(\mathbb R^m)$ satisfying the support condition (1.11), that the closure of the support of $w$ avoids the coordinate hyperplanes $x_1\cdots x_m=0$.

$E_{F,w}(X)$ and (1.11) are defined on the Theorem 1.6 page; Conjectures 1.2 and 1.5 are stated on the Theorem 1.3 page. The two new hypotheses, as the paper states them:

• Conjecture 1.10 (RA1$\delta$) (p. 6). For even $m$, assuming Conjecture 1.2, the effective form of Conjecture 1.8: there is a real $\eta_1=\eta_1(F)>0$, depending only on $F$, such that for $Z\ge2$, $M\in[1,Z^{\eta_1}]$, a modulus $n_0\in[1,M]$, $\mathbf a,\mathbf b\in\mathbb Z^m\cap[-M,M]^m$, $t\in[-M,M]$ and $s=\sigma(Z)+it$ the sum of $\Phi^{\mathbf c,1}(s)$ over $\mathbf c\in\mathcal S_1\cap Z\cdot\mathcal B_M(\mathbf b)$ with $\mathbf c\equiv\mathbf a\bmod n_0$ equals the sum over the same $\mathbf c$ of $(1+O_F(Z^{-\eta_1}))A^{\mathbf a,n_0}_{F,1}(s)$ (display (1.15)).
• Conjecture 1.11 (EKL) (p. 6). There is a nonzero homogeneous polynomial $H\in\mathbb Z[\mathbf c]$ with $H/\Delta\in\mathbb Z[\mathbf c]$ such that for all primes $p$ and all $\mathbf a,\mathbf b\in\mathbb Z_p^m$ with $H(\mathbf b)\ne0$ and $\mathbf a\equiv\mathbf b\bmod pH(\mathbf b)$, one has $H(\mathbf a)\ne0$ and $L_p(s,V_{\mathbf a})=L_p(s,V_{\mathbf b})$: an effective local constancy of the local factor.

All four hypotheses are unproved, so the power saving is conditional. The paper notes (p. 6) that a version uniform over small perturbations would, with its reference [Wan23c], give that 100% of primes $p\not\equiv\pm4\pmod9$ are sums of three integer cubes under the same hypotheses; it does not carry this out.

Proof pointer

p. 58. The print says to proceed "as in the proof of Theorem 10.7" [sic], evidently meaning the proof of Theorem 1.6, which starts from (10.30), with Theorem 10.8 (p. 57) in place of Theorem 10.7; this gives $E_{F,w}(X)/X^3\ll_\epsilon X^{-0.25+\epsilon}+X^{-\eta_{12}}$, with $\eta_{12}=\eta_{12}(F)>0$ the constant of Theorem 10.8.

Dependencies

Conjectures 1.2, 1.5, 1.10 and 1.11 as hypotheses; within the paper, Theorem 2.5, Propositions 9.7 and 9.9, and Theorem 10.8.

Bears on

No Erdős problem directly; the paper's result bearing on Problem 940 is Theorem 1.3.