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Source. Pipeline-math, Erdős problem 477, commit 99d916ff32a90e77c98eb004537ccda409262346 (29 June 2026), Proposition 1.6, printed/PDF pp. 4-5 of the manuscript. The shell-to-box deduction below is a local reconstruction step absent from the manuscript's proof. It uses the cited 2009 journal version of Heath-Brown's Theorem 2.

Statement

Put $B=\{m^{13}:m\in\mathbb Z\}$ and $D=B-B$. For a fixed $c\in\mathbb Z\setminus B$ and real $T\ge1$, define

$$S_c(T)=\{t\in\mathbb Z:|t|\le T,\ t^{13}-c\in D\}.$$

Then

$$|S_c(T)|=O_c(T^{5/6}),\qquad \lim_{T\to\infty}\frac{|S_c(T)|}{T}=0.$$

The implied constant may depend on $c$ and is independent of $T$. No uniform bound over all integer shifts is asserted.

External premise and version

We use Heath-Brown's Theorem 2, Journal of Number Theory 129 (2009), printed p. 1580, PDF p. 2, in the journal version of record. For a nonsingular integral ternary form $F$ of degree $k\ge3$ and a positive integer $N\ll_F R$, it gives

$$\#\{\mathbf x\in\mathbb Z^3:F(\mathbf x)=N, R/2<\|\mathbf x\|_\infty\le R,\ \mathbf x\notin S_{\lfloor k/10\rfloor}\} \ll_F R^{10/k}. \tag{1}$$

Here $S_d$ consists of solutions in polynomial families $\mathbf f(s)\in\mathbb Z[s]^3$ with $F(\mathbf f(s))\equiv N$ and positive maximum degree at most $d$. The introductory definition does not explicitly exclude degree-zero triples. The positive-degree, nonconstant convention is an inference from the source's discussion of parametrized curves and its $O(R^{1/d})$ family count, not an additional hypothesis explicitly printed there. The reconstruction author visually read this family count on journal printed p. 1589 (PDF p. 11), in Section 4, and the corresponding context on arXiv v1 pp. 10-11. The journal passage treats parametric families and uses $O(R^{1/d})$; the positive-degree convention is inferred from that context, not introduced as an explicit source definition. Our exclusion of all nonconstant rational families makes the point-evaluation convention immaterial to this application. We do not regard individual constant triples as exceptional families.

The journal definition in (1) counts a shell. In contrast, arXiv:0806.4330v1 defines its count on p. 1 using $\|\mathbf x\|_\infty\le R$. The manuscript states a whole-box version as Theorem 1.3 while citing the journal, with a constant depending only on $F$. The journal theorem does not provide that uniform whole-box statement. We use only the journal shell statement and the fixed-$c$ box deduction proved below, whose constant may depend on $c$. The manuscript's Proposition 1.6 proof has no shell summation, and so no bound for the leftover box below the shells: it applies Theorem 1.3 to the whole box for large $X$, and on p. 5 it lets the implied constant cover the bounded range of smaller $X$. Our distinction between the journal and v1 counting definitions is also separate from the manuscript's whole-box restatement. This is not an author-issued erratum. Heath-Brown's proof remains an external literature premise.

Proof

Bound every witnessing pair

If $t\in S_c(T)$, there are integers $u,v$ with

$$t^{13}-c=u^{13}-v^{13}. \tag{2}$$

The equality $u=v$ would force $c=t^{13}\in B$, so $u\ne v$. Let

$$Q(u,v)=\sum_{j=0}^{12}u^{12-j}v^j, \qquad u^{13}-v^{13}=(u-v)Q(u,v).$$

For distinct real $u,v$, the quotient $(u^{13}-v^{13})/(u-v)$ is positive because the odd power map is strictly increasing. On the diagonal away from the origin, $Q(u,u)=13u^{12}>0$. Thus the continuous homogeneous degree-12 polynomial $Q$ is positive on the compact set $\max(|u|,|v|)=1$. Its minimum there is some $\kappa>0$. Scaling gives

$$Q(u,v)\ge\kappa\max(|u|,|v|)^{12} \quad\text{for all }(u,v)\in\mathbb R^2,$$

including the origin. Since $u,v$ are distinct integers, $|u-v|\ge1$. Equation (2) and $T\ge1$ therefore imply

$$\kappa\max(|u|,|v|)^{12} \le |u^{13}-v^{13}| =|t^{13}-c| \le(1+|c|)T^{13}.$$

Consequently $\max(|u|,|v|)\le C_cT^{13/12}$ for a constant $C_c\ge1$ depending only on $c$. This bound holds for every pair witnessing (2).

Remove the exceptional families

Set $(x,y,z)=(u,-v,-t)$ and $X=C_cT^{13/12}$. Since $T\le T^{13/12}$, the resulting triple satisfies

$$x^{13}+y^{13}+z^{13}=-c,\qquad \max(|x|,|y|,|z|)\le X. \tag{3}$$

As $c\notin B$ and $0\in B$, $c\ne0$. Put

$$\epsilon_c=\operatorname{sgn}(-c),\qquad F_c(X_1,X_2,X_3)=\epsilon_c(X_1^{13}+X_2^{13}+X_3^{13}), \qquad N_c=|c|>0.$$

Equation (3) is equivalent to $F_c(x,y,z)=N_c$. This is an integral ternary form of degree 13. Its three first derivatives are nonzero constant multiples of $X_i^{12}$, so their only simultaneous zero over $\overline{\mathbb Q}$ is the origin. It is therefore nonsingular as a projective ternary form.

The excluded degree threshold in (1) is $\lfloor13/10\rfloor=1$. A nonconstant polynomial identity $F_c(\mathbf f(s))=N_c$ would, after changing the signs of the second and third coordinates, give a nonconstant rational curve on $u^{13}-v^{13}-t^{13}=-c$. This is impossible by Corollary 1.5. Thus the exceptional family set is empty, and (1) counts all solutions of (3) in each admissible shell.

Sum the journal shells for a fixed shift

Keep $c$ fixed. Choose a threshold $R_c\ge1$ large enough that $N_c\ll_{F_c}R$ is in the theorem's range whenever $R\ge R_c$. This is possible because $N_c$ is fixed. The corresponding estimate in (1) is uniform as the shell height $R$ varies above $R_c$.

If $X\ge R_c$, put $R_j=X/2^j$, and choose the largest integer $J\ge0$ for which $R_J\ge R_c$. The shells

$$R_j/2<\|\mathbf x\|_\infty\le R_j, \qquad 0\le j\le J,$$

are disjoint and cover the part of the box above $R_{J+1}<R_c$. By (1), their total contribution is at most

$$K_{F_c}\sum_{j=0}^{J}(X/2^j)^{10/13} \le\frac{K_{F_c}}{1-2^{-10/13}}X^{10/13}. \tag{4}$$

The leftover box has height less than $R_c$ and contains at most $(2\lceil R_c\rceil+1)^3$ integer triples, whether or not they satisfy the equation. This is a constant depending on $c$. When $1\le X<R_c$, the same fixed bound covers the entire box. Since $X^{10/13}\ge1$, (4) and this bounded contribution give, for all $X\ge1$,

$$\#\{\mathbf x\in\mathbb Z^3:F_c(\mathbf x)=N_c, \|\mathbf x\|_\infty\le X\}=O_c(X^{10/13}). \tag{5}$$

This argument does not apply the theorem at scales where its range condition fails. The threshold and bounded remainder may depend on $c$; (5) is not a uniform-in-$c$ whole-box claim.

Count bad parameters

Every $t\in S_c(T)$ has at least one triple in (3). Projection of these triples to $-z$ therefore covers $S_c(T)$. Different parameters have different third coordinates, so the number of parameters is no larger than the number of triples. Applying (5) yields

$$|S_c(T)|\ll_c(C_cT^{13/12})^{10/13} \ll_c T^{5/6}.$$

Dividing by $T$ gives a bound by a fixed multiple of $T^{-1/6}$, which tends to zero. This proves both assertions.

Dependencies and current verification

The reconstruction consumes Corollary 1.5 and the journal Theorem 2 under its contextually inferred positive-degree-family convention. The reconstruction author visually checked the journal statement and definitions at printed p. 1580, arXiv v1 pp. 1-2 for the version difference, and journal p. 1589 (PDF p. 11) and v1 pp. 10-11 for family terminology. Journal pp. 1580 and 1589 were also read in extracted text. These were complete-page visual readings, with the later proof passages read only for context. They do not reconstruct Heath-Brown's determinant-method proof. The shell summation and leftover-box bound above are local reconstruction steps absent from the manuscript's Proposition 1.6 proof, not an author-issued erratum.

This complete reconstruction of Proposition 1.6 received [[diophantine_problems/pipeline_math_2026_tiling_complement/evidence/verify/compilation_review|independent compilation review]]. No material defect was found in its exact frozen statement, essential deductions or consumed interfaces, including the fixed-shift shell summation and finite small-scale bound. The manuscript's pp. 4-5 were read in text and rendered images. Attack selection was partly pre-directed; the derivations were independently performed. The six-result review is relative to the Corvaja-Zannier-recalled unit bounds and Heath-Brown's journal Theorem 2, with the recorded nonconstant-family qualification. The external proofs were not independently reviewed; no formal verification is claimed. Source versions are recorded in the [[diophantine_problems/pipeline_math_2026_tiling_complement/_index|source digest]].

Bears on. The fixed-shift estimate is used by Proposition 1.8, then by Problem 477.