Abstract
Let A be an infinite set of nonnegative integers. For h ≥ 2, let hA be the set of all sums of h not necessarily distinct elements of A. Suppose that l ≥ 2, and that every sufficiently large integer in the sumset hA has at least l representations. If l = 2, then \(A(x) \geq (\log x/\log h) - w_{0}\), where A(x) counts the number of integers a ∈ A such that 1 ≤ a ≤ x. Lower bounds for A(x) are also obtained for l ≥ 3.
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