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 $A_n = P_nG_n \subset F$, 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.