Long arithmetic progressions in sets with small sumsets

This paper continues the series of papers on Inverse Additive Number Theory published in 1955–1964 (see references [84]–[92], [98] in [2]). Throughout the paper, we work with the set A ⊂ Z of cardinality |A| = k ≥ 3. We assume that A = {a0 = 0 < a1 < · · · < ak−1} and that the greatest common divisor of the numbers from A is 1. Let T denote the cardinality of the set 2A = A + A of all pairwise sums a + b of numbers from A. Notice that T ≥ 2k − 1. In [1] (see also the textbook [3, p. 204]), we proved the following result. Theorem 1. For 0 ≤ b < k−2 and T = 2k−1+b, the set A is contained in (1) L = {0, 1, 2, . . . , k + b− 1}.