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What is the structure of a pair of finite integers sets A,B ⊂ ℤ with the small value of |A+B|? We answer this question for addition coefficient 3. The obtained theorem sharpens the corresponding results of G. Freiman.
0. Introduction. Since ℤ is a principal ideal domain, every finitely generated torsion-free ℤ-module has a finite ℤ-basis; in particular, any fractional ideal in a number field has an "integral basis". However, if K is an arbitrary number field the ring of integers, A, of K is a Dedekind domain but not necessarily a principal ideal domain. If L/K is a finite extension of number fields, then...
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