The sum-product conjecture holds for fixed-field lattice product constructions
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 An = PnGn ⊂ F, where F is a fixed number field, Pn can be rescaled to a set of algebraic integers whose conjugates are polynomially bounded in |Pn ∖ {0}|, and Gn occupies a positive proportion of a bounded box of scale Yn in a fixed positive-rank multiplicative lattice. We prove that these fixed-field constructions satisfy the sum-product conjecture, max{|An+An|, |AnAn|} ≥ |An|2−o(1) as |An| → ∞, 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∈ (0,1], if |AnAn| ≤ |An|2−δ for all sufficiently large n, then Yn → ∞ and |An+An|/|An|2 = 1/2 + Oδ((log Yn)C/Yn). 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.
Related
BibTeX
@misc{lemley2026sumproduct,
title = {{The sum-product conjecture holds for fixed-field lattice product constructions}},
author = {Harrison Lemley and David Chew},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26737},
eprint = {2610.26737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26737},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The sum-product conjecture holds for fixed-field lattice product constructions
AU - Harrison Lemley
AU - David Chew
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.26737
UR - https://arxiv.org/abs/2610.26737
AB - 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 ⊂ 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 ∖ {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|} ≥ |A_n|^2−o(1) as |A_n| → ∞, 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∈ (0,1], if |A_nA_n| ≤ |A_n|^2−δ for all sufficiently large n, then Y_n → ∞ 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.
ER -