Near-quadratic sum-product estimates in fixed number fields

Let $K$ be a fixed number field. We prove a quantitative expansion theorem for mixed sumsets $A+B$, assuming only that $A$ is covered by relatively few cosets of a multiplicative group of small finite valuation rank. The constants are independent of the generators, their heights, and their prime ideal supports. As a consequence, if $A\subset K^\times$, $|A|=N\ge 3$, and each element has nonzero valuation at at most $k$ finite places, then every $B\subset K$ with $|B|=N$ satisfies $$ \max\{|A+B|,|AA|\} \ge N^2\exp\left[-C_K\left((k+1)\log(\log N+2)+\sqrt{(k+1)\log N}\right)\right]. $$ This is $N^{2-o(1)}$ for $k=o(\log N/\log\log N)$. We also obtain a trade-off between the additive and multiplicative deficits, estimates for iterated linear sumsets, and a transfer theorem for product constructions $A=PG$ in terms of the multiplicative deficit of $P$. The proof recovers local scales from an exact sum and bounds the remaining conditional entropy by the overlap of entropy measures on fixed trees.