Beyer de Ryke: A density deficit for sums of three cube-full numbers

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Basile Beyer de Ryke, "A density deficit for sums of three cube-full numbers," unpublished manuscript, revised 26 July 2026. The copy read for this card is arXiv's submission-preview build of the paper, stamped "arXiv:submit/7835508 [math.NT] 26 Jul 2026" and built seven minutes before the v1 submission time, announced as arXiv:2609.35772 (v1 of 26 July 2026 is the only version, with the same title and author); the arXiv record names arXiv's non-exclusive distribution license (https://arxiv.org/abs/2609.35772, read 2026-10-02), every other right reserved.

Overview

The manuscript studies the additive set $\mathcal R_3=\mathcal F+\mathcal F+\mathcal F$, where $\mathcal F$ consists of positive cube-full integers and includes $1$. Its main result is not a density-zero theorem: Theorem 1.1 (p. 1; proof pp. 6–7) gives, for each $\varepsilon>0$, a modulus $M\geq1$ and a residue $a$ coprime to $M$ such that the upper density of $\mathcal R_3$ relative to the progression $a\pmod M$ is at most $\varepsilon$, namely

$$\limsup_{X\to\infty}\frac{M}{X}\mathcal R_3(X;a,M)\leq\varepsilon.$$

Corollary 1.2 (p. 1) consequently gives positive lower natural density for $\mathbb N\setminus\mathcal R_3$. Equation (3) (p. 3) implies $\#(\mathcal F\cap[1,X])=O(X^{1/3})$, so values represented with one or two summands contribute only $O(X^{2/3})$; hence the complement of the integers representable by at most three cube-full numbers also has positive lower density. The paper records as open whether three-term sums have positive density (Section 1, p. 2); it proves neither that nor density zero.

The structural input is the unique canonical factorization

$$n=x^3b^4c^5,$$

with $bc$ squarefree (Lemma 2.1, p. 3). Thus every cube-full number is uniquely $dx^3$, where $d=b^4c^5$ belongs to the set $\mathcal C$ of canonical 4-full parts. The weighted sums

$$W=\sum_{d\in\mathcal C}d^{-1/3},\qquad H=\sum_{d\in\mathcal C}3^{\omega(d)}d^{-1/3}$$

converge by the Euler products (1) and (2) (p. 3). The second, stronger weight absorbs primes dividing uncontrolled 4-full parts.

The local saving comes from cubic characters. For $q\equiv1\pmod3$, Lemma 3.1 (pp. 3–4) evaluates the number $N_q(u)$ of solutions of $x^3+y^3+z^3=u$ exactly in (4), using the cubic exponential-sum identity (6). If $u$ is a non-cube, (5) gives $N_q(u)/q^2\leq1-2/q$. Lemma 3.2 (p. 4) supplies the uniform fallback estimate (7) for arbitrary coefficients, with a factor $3$ for each coefficient divisible by $q$.

For fixed 4-full parts $\mathbf d=(d_1,d_2,d_3)$, Proposition 4.1 (p. 5), equation (8), combines the Chinese remainder theorem with Davenport’s lattice-point principle to obtain

$$R_{\mathbf d}(X;a,M)=\frac{VX}{M}(d_1d_2d_3)^{-1/3}\prod_{q\mid M}\rho_q(a;\mathbf d)+O_{M,\mathbf d}(X^{2/3}),$$

where $V=\Gamma(4/3)^3$. The dependence of the error term on $\mathbf d$ is why the proof must truncate the canonical parts before invoking this proposition.

Given a finite set $D$ of canonical parts, Lemma 5.1 (pp. 5–6) shows that primes splitting completely in $\mathbb Q(\zeta_3,\sqrt[3]{d}:d\in D)$, apart from finitely many excluded primes, satisfy $q\equiv1\pmod3$ and make every $d\in D$ a nonzero cube modulo $q$. The use of positive-density splitting primes is an application of the cited Chebotarev density theorem, not a result proved in the manuscript. In the proof of Theorem 1.1, $D$ is first chosen through the $H$-tail bound (10); finitely many splitting primes are then selected with reciprocal sum in the interval (11), and non-cubes $u_q$ are combined into a reduced class $a\pmod M$. A larger finite set $E$ is chosen through the $W$-tail condition (12). Contributions from $D^3$, from $E^3\setminus D^3$, and from triples outside $E^3$ are bounded respectively in (13), (14), and (15) (pp. 6–7). Their sum proves the asserted relative-density deficit.

Section 6 explains the scope limitation. The general canonical decomposition for $r$-full integers is (16), and its summable weight is (17) (p. 7). Proposition 6.1 (p. 8) proves that the number of ordered $r$-tuples of positive $r$-full integers with total at most $X$ is asymptotic to $\Gamma(1+1/r)^rW_r^rX$, with the corresponding unordered constant divided by $r!$. Thus elementary tuple counting has order $X$, not $o(X)$, and the manuscript states that its cubic local-restriction argument does not give an equally effective replacement when $r\geq4$.

Relation to E940

This source bears on Problem 940.

In E940’s notation, let $\mathcal F_r$ be the positive $r$-powerful (the paper says $r$-full) integers and

$$S_{\le r}=\bigcup_{k=1}^{r}(\underbrace{\mathcal F_r+\cdots+\mathcal F_r}_{k\text{ terms}}).$$

For $r=3$, the paper’s $\mathcal F$ is $\mathcal F_3$ and its $\mathcal R_3$ is the exactly-three-summand set $3\mathcal F_3$. Theorem 1.1 provides, for every $\varepsilon>0$, a modulus and reduced class on which $3\mathcal F_3$ occupies at most an $\varepsilon$-fraction asymptotically. Corollary 1.2 together with (3) yields

$$\underline d(\mathbb N\setminus S_{\le3})>0.$$

The manuscript states that this proves the infinitude assertion of E940 for $r=3$ (abstract, p. 1), in fact with a positive-density exceptional set; the manuscript is unrefereed, its proof was not reviewed for this card, and the claim's standing is recorded on its claim page.

The result does not prove E940’s requested conclusion $d(S_{\le3})=0$. The modulus $M$ depends on $\varepsilon$, and sparsity inside one progression—even with arbitrarily small relative density after changing the progression—does not imply global density zero. It is also compatible with $S_{\le3}$ having positive density; Section 1 (p. 2) records that this is unknown.

The constructions most directly reusable in work on E940 are the canonical decomposition (16) and convergent weight (17), which apply to every $r$, and the finite-core/two-tail architecture of (10)–(15). For $r=3$, Lemma 3.1 supplies the decisive local factor $1-2/q$, while Lemma 3.2 controls parts outside the finite core. An extension to $r\ge4$ would need an analogue producing sufficiently strong multiplicative local savings while retaining summable tail weights. Proposition 6.1 shows why counting representations alone cannot establish density zero: there are asymptotically a positive constant times $X$ candidate $r$-tuples. The manuscript proves no density deficit, infinitude result, or density-zero statement for any $r\ge4$, and it does not resolve the density-zero question for $r=3$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.