## Abstract

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 *h _{n}* of positive integers such that the relative density of

*A*in [0,

*h*] approaches α, the set

_{n}*A*∩ [0,

*h*] can be partitioned into two parts

_{n}*A*∩ [0,

*c*] and

_{n}*A*∩ [

*b*,

_{n}*h*], such that

_{n}*c*approaches 0, i.e. the size of

_{n}/h_{n}*A*∩ [0,

*c*] is asymptotically small compared with the size of [0,

_{n}*h*], and (

_{n}*h*−

_{n}*b*)/

_{n}*h*approaches α, i.e. the size of A ∩ [

_{n}*b*,

_{n}*h*] is asymptotically almost the same as the size of the interval [

_{n}*b*,

_{n}*h*]. The results here answer a question of the author in [

_{n}*R. Jin*, Inverse problem for upper asymptotic density, Trans. Amer. Math. Soc.

**355**(2003), No. 1, 57–78.]

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