Abstract
Let A be a set of nonnegative integers. The h-fold sum of A, denoted hA, is the set consisting of all sums of h not necessarily distinct elements of A. The set A is an asymptotic basis of order if hA contains all sufficiently large integers. The set A is an asymptotic basis if A is an asymptotic basis of order h for some hel. If A is an asymptotic basis, the exact order of A, denoted g(A), is the smallest integer h such that A is an asymptotic basis of order h. Let kel. l'f A is an asymptotic basis, let Ik(A) denote the set of all subsets F~-A such that F has cardinality k and the set A\F is an asymptotic basis. An open problem in additive number theory is to estimate g(A\F) in terms of g(A). More precisely, define
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