Lev, Vsevolod F.; Smeliansky, Pavel Y. On addition of two distinct sets of integers. (English) Zbl 0817.11005 Acta Arith. 70, No. 1, 85-91 (1995). 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) Cited in 6 ReviewsCited in 32 Documents MSC: 11B13 Additive bases, including sumsets 11B83 Special sequences and polynomials Keywords:set addition; doubling; inverse problems; additive number theory; addition of distinct sets × Cite Format Result Cite Review PDF Full Text: DOI EuDML