Infinite Sumsets with Many Representations

Melvyn B. Nathanson ()
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Melvyn B. Nathanson: Lehman College (CUNY), Department of Mathematics

A chapter in Analytic Number Theory, 2015, pp 275-283 from Springer

Abstract: 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 ) ≥ ( log x ∕ log h ) − w 0 $$A(x) \geq (\log x/\log h) - w_{0}$$ , 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.

Keywords: Sumset; Nonnegative Integer; Counting Function; Balasubramanian; Information Theory (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-22240-0_16

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