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.
To Helmut Maier on his 60th birthday
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References
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Acknowledgements
I thank Michael Filaseta for bringing these problems to my attention, and Quan-Hui Yang for the reference to the paper of Balasubramanian and Prakesh.
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Nathanson, M.B. (2015). Infinite Sumsets with Many Representations. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_16
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DOI: https://doi.org/10.1007/978-3-319-22240-0_16
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