Result: A generalization of Freiman's 3k-3 Theorem: A generalization of Freiman's \(3k-3\) theorem

Let \(A\) be a set of integers, \(|A|=k\). A classical result of Freiman asserts that if \(|A+A|\leq 3k-4\), then \(A\) is contained in a short arithmetic progression, while for \(|A+A|=3k-3\) there is another structure (two progressions) and an isolated case. This theorem is now extended to sums of the form \(A+tA\), where \(tA = \{ta: a\in A \}\). The conclusion is the same: either \(A\) is contained in a short arithmetic progression, or \(t=1\) or \(-1\) and \(A\) is the union of two arithmetic progressions with a common difference, or \(A\) is the same 6-element exceptional configuration. The proof is based on the isoperimetric method of the first author.