Solution to the inverse problem for upper asymptotic density

Inverse problems study the structure of a set A when the “size” of A + A is small. In the article, the structure of an infinite set A of natural numbers with positive upper asymptotic density is characterized when A is not a subset of an infinite arithmetic progression of difference greater than one and A + A has the least possible upper asymptotic density. For example, if the upper asymptotic density α of A is strictly between 0 and 1/2, the upper asymptotic density of A + A is equal to 3α/2, and A is not a subset of an infinite arithmetic progression of difference greater than one, then A is either a large subset of the union of two infinite arithmetic progressions with the same common difference k = 2/α or for every increasing sequence hn of positive integers such that the relative density of A in [0, hn ] approaches α, the set A ∩ [0, hn ] can be partitioned into two parts A ∩ [0, cn ] and A ∩ [bn , hn ], such that cn/hn approaches 0, i.e. the size of A ∩ [0, cn ] is asymptotically small compared with the size of [0, hn ], and (hn − bn )/hn approaches α, i.e. the size of A ∩ [bn , hn ] is asymptotically almost the same as the size of the interval [bn , hn ]. The results here answer a question of the author in [R. Jin, Inverse problem for upper asymptotic density, Trans. Amer. Math. Soc. 355 (2003), No. 1, 57–78.]