Let G be a group on which a measure m is defined. If A, B C Gwe define A © B = C = {c\c = a + b, a E A, b S b}. By Ak C G we denote a subset of G consisting of k elements. Given Ak we define s(Ak) = inf m {b\B C G, Ak @ B = G} and ck = supA CGs(Ak). Theorems 1, 2, and 3 deal with the problem of determining ck. In the dual problem we are given B, m(B) > 0, and required to find minimal A such that A © B = G or, sometimes, m(A © B) = m{G). Theorems 5 and 6 deal with this problem. Let A and B be

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... s of nonnegative integers, with 0 G A The set B is called a complement of A if each nonnegative integer is expressible in the form a + b ia G A, b G B). One of the basic problems in additive number theory is the determination, for a prescribed A, of a complement B that is in some sense minimal. Erdös [1] and Lorentz [2] have discussed some problems and concepts for the case where A is an infinite set; D. J. Newman [3] has dealt with finite sets A. We have also obtained some results for the case where A is finite, and they will appear elsewhere [5] . Here we generalize this concept in several respects. Let G be a group on which a measure m is defined. If A, B C G we define A ® B = {c\c =a + b,aGA,bGB}. By Ak C G we denote a subset of G