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On addition of two distinct sets of integers. (English) Zbl 0817.11005

A number of results due to G. Freiman generalizing his own “\((3k-3)\)- theorem” for the case of addition of different sets are sharpened, given a complete form and a new short proof. One of our central results is the following.
Let \(A= \{a_ 1,\dots, a_ k\}\) and \(B= \{b_ 1, \dots, b_ l\}\) be two sets of integers, where \(0= a_ 1< \dots< a_ k\), \(0= b_ 1< \dots <b_ l\), \(b_ l\leq a_ k\), and put \[ \delta= \begin{cases} 0, \quad &\text{if } b_ l< a_ k\\ 1, \quad &\text{if } b_ l =a_ k. \end{cases} \] We have:
i) If \(a_ k\leq k+l- 2-\delta\), then \(| A+ B|\geq a_ k+l\).
ii) If \(a_ k\geq k+l - 1 -\delta\), and \(\text{gcd} (A)=1\), then \(| A+ B|\geq k+2l- 2-\delta\).
Reviewer: V.F.Lev (Tel-Aviv)

MSC:

11B13 Additive bases, including sumsets
11B83 Special sequences and polynomials