arXiv:2610.26737v1

The sum-product conjecture holds for fixed-field lattice product constructions

math.NTmath.CO11B3011J8611R2711G5011P70

Paper metadata

Original submission: October 19, 2026
Claimed MSC: 11B30, 11J86, 11R27, 11G50, 11P70
arXiv categories: math.NT, math.CO
Paper license: CC BY 4.0 ↗
Comments: 25 pages

Abstract

We study fixed-field analogues of the lattice product construction recently used by Bloom, Sawin, Schildkraut, and Zhelezov to disprove the sum-product conjecture over the real numbers. These sets are of the form $A_n = P_nG_n \subset F$ , where $F$ is a fixed number field, $P_n$ can be rescaled to a set of algebraic integers whose conjugates are polynomially bounded in $|P_n \setminus \{0\}|$ , and $G_n$ occupies a positive proportion of a bounded box of scale $Y_n$ in a fixed positive-rank multiplicative lattice. We prove that these fixed-field constructions satisfy the sum-product conjecture, $\max\{|A_n+A_n|, |A_nA_n|\} \ge |A_n|^{2-o(1)}$ as $|A_n| \to \infty$ , with no restriction on the relative growth of the two factors. More strongly, any fixed power saving in the product set forces an asymptotically maximal sumset: there exists $C > 0$ such that, for each fixed $δ\in (0,1]$ , if $|A_nA_n| \le |A_n|^{2-δ}$ for all sufficiently large $n$ , then $Y_n \to \infty$ and $|A_n+A_n|/|A_n|^2 = 1/2 + O_δ((\log Y_n)^C/Y_n)$ . In particular, the additive-box times unit-box construction of Bloom, Sawin, Schildkraut, and Zhelezov cannot produce sum-product counterexamples in any fixed totally real number field, at any relative scales. The proof combines a bound for factorizations of algebraic integers with a more general maximal-sumset theorem for coefficient sets whose nonzero-element ratios have small Weil height. The latter controls multiplication fibers and combines a lattice strip estimate with logarithmic separation derived from the theorems of Yu and Matveev.

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Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated October 23, 2026

01Statements3 reported findingsNo statement issues found

The fixed-field obstruction and the quantitative maximal-sumset theorem are supported. Product saving forces the multiplicative scale to grow and the ratio heights to be logarithmic in that scale. Two local formula corrections are recorded without changing these conclusions.

Theorem 3.1; Proposition 3.8No issue found

Logarithmic separation yields the quantitative maximal-sumset estimate

PDF pp. 10–18 · arXiv:2610.26737v1

At a fixed nontrivial place, a lattice strip of width $H$ contains $O(Y^{\rho-1}(H+1))$ exponent points. The logarithmic lower bound for a nontrivial ratio gives separation at width $H\asymp(\log Y)^{\rho+4}$ . Two good unordered representations of a sum must then coincide, so the collision excess is at most $O(M^2Y^{2\rho-1}(\log Y)^{\rho+4})$ . Height-counting bounds the multiplication fibers by $O((\log Y)^\rho)$ , and division by $|A|^2$ gives the asserted error with $C=3\rho+4$ . The possible zero element contributes only a lower-order term.

Theorem 1.3; Lemmas 4.1–4.4No issue found

A fixed product saving forces the required scale regime

PDF pp. 18–21 · arXiv:2610.26737v1

The ideal-divisor estimate, followed by bounded-height counting in the fixed unit group, gives $|P^*P^*|\gg_\varepsilon M^{2-\varepsilon}$ . Comparing this with $|AA|\leq |A|^{2-\delta}$ and $|A|\leq2M|G|$ gives $M^{\delta/2}\ll_\delta |G|^{2-\delta}$ . Hence $M$ is polynomially bounded in $Y$ , $R(P)\ll_\delta\log Y$ , and $|A|\to\infty$ forces $Y\to\infty$ . Applying Theorem 3.1 on any purported counterexample subsequence contradicts a simultaneous fixed power saving in the sumset. Constants depend only on the stated fixed data and, where indicated, on $\delta$ .

Corollaries 1.4 and 5.2No issue found

Both applications satisfy the stated fixed-field hypotheses

PDF pp. 21–24 · arXiv:2610.26737v1

The Minkowski lattice gives coefficient cardinality comparable to $X^d$ , and its conjugates are $O(X)$ . The one-sided unit-log box is a bounded full-dimensional polytope in the sum-zero hyperplane, so it has $\asymp Y^{d-1}$ lattice points and exponent coordinates $O(Y)$ . This proves the unit-family admissibility after the local range correction recorded below. For arithmetic–geometric products, independent exponent vectors give at least $n^\rho$ distinct group elements, while clearing fixed denominators makes every nonzero coefficient ratio have height $O(\log n)$ .

02Proofs3 reported findingsNo proof issues found

The separation, collision, divisor-counting, and scale-forcing arguments close. The displayed unit-box domain needs a local range correction, and the quotation of one fixed prime-dependent height threshold has a harmless misplaced square.

Corollary 3.5; Lemma 3.6No issue found in the reviewed proof

The logarithmic separation bound is uniform in the varying bases

PDF pp. 11–15 · arXiv:2610.26737v1

Appending a fixed primitive field generator with zero exponent makes the finite-place theorem apply over the fixed field. All height parameters are bounded by a sufficiently large multiple of $\log Y$ , and the exponent parameters by a multiple of $Y$ . The product of height parameters and the two remaining logarithmic factors give the claimed coarse exponent $s+3$ . At an archimedean place the principal logarithm adds an integer multiple of $2\pi i$ of size $O(Y)$ ; Matveev's estimate and $|\operatorname{Log}z|\leq2|z-1|$ give the required lower bound. No independence of the logarithms is assumed.

Lemma 5.1Suspected minor formal issue

Use the one-sided unit-log polytope in the exact description of the unit box

PDF p. 22 · definition of the polytope and the exact display for the unit set · arXiv:2610.26737v1

The unit box on p. 4 imposes only $\log|\sigma_i(u)|\leq Y$ , whereas p. 22 identifies it with $\|M_\times x\|_\infty\leq Y$ , which also imposes lower bounds. Replace the displayed polytope by $\Omega_\times=\{x:(M_\times x)_i\leq1\text{ for all }i\}$ . Every image vector has coordinate sum zero, so these upper bounds immediately imply lower bounds $-(d-1)$ ; the polytope is bounded and contains a neighborhood of zero. The same two-cube count and exponent bound therefore apply without a new estimate. The cardinality, admissibility, and corollary are unchanged.

Quotation of Yu's height conditionSuspected typo

The fixed factor log p lies outside the square in Yu's condition

PDF p. 12 · display quoting condition (0.21)* · arXiv:2610.26737v1

Yu's printed condition is $V_j\geq\max\{h(\alpha_j),|\log\alpha_j|/(10D_0),\tfrac12(f_0/D_0)^2\log p\}$ , with $\log p$ outside the square. The manuscript places it inside the square. This is a fixed prime-dependent threshold, and the proof already chooses the coefficient in $V_*=C_1\log Y$ sufficiently large depending on the fixed place. Correcting the quotation changes only that fixed coefficient and leaves the bound and all later arguments unchanged.

Kunrui Yu, Linear forms in p-adic logarithms III, printed p. 246, condition (0.21)* and Theorem 1′ ↗

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2610.26737v1
Authors listed: Harrison Lemley, David Chew
Audit date: October 23, 2026

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The sum-product conjecture holds for fixed-field lattice product constructions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Logarithmic separation yields the quantitative maximal-sumset estimate

A fixed product saving forces the required scale regime

Both applications satisfy the stated fixed-field hypotheses

The logarithmic separation bound is uniform in the varying bases

Use the one-sided unit-log polytope in the exact description of the unit box

The fixed factor log p lies outside the square in Yu's condition

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