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⊂ K^×, |A|=N≥ 3, and each element has nonzero valuation at at most k finite places, then every B⊂ K with $|B|=N$ satisfies max\|A+B|,|AA|\ ≥ N²exp[-CK((k+1)log(log N+2)+√(k+1)log N)]. This is N²⁻ᵒ⁽¹⁾ for k=o(log N/loglog 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.