Inverse Problem For Upper Asymptotic Density II

Inverse problems study the structure of a set A when the A+A is “small”. In the article, the structure of an infinite set A of natural numbers is described when A+A has the least possible upper asymptotic density and A contains two consecutive numbers. For example, if the upper asymptotic density α of A is between 0 and 2 , the upper asymptotic density of A+A is less than or equal to 3 2α, and A contains two consecutive numbers, then A is either a large subset of the union of two arithmetic sequences with same common difference k = 2 α , or for any 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 cardinality of A ∩ [0, cn] is relatively very small, and (hn − bn)/hn approaches to α, i.e. the cardinality of A∩ [bn, hn] is relatively the same as the cardinality of the interval [bn, hn].