2003/05/06 by Melvyn B. Nathanson, Nathanson, Melvyn B.
Mathematics · #05A30 #11B05 #11B13 #11B34 #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A30 #msc:11B05 #msc:11B13 #msc:11B34
paper · pdf · doi:10.48550/arxiv.math/0305087
10 pages, LaTex
arxiv created 2003/05/06 · openalex publication_date 2003/05/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a set of integers. For every integer n, let rA,2(n) denote the number of representations of n in the form n = a1 + a2, where a1 and a2 are in A and a1 ≤ a2. The function rA,2: Z → N0 ∪ ∞ is the representation function of order 2 for A. The set A is called an asymptotic basis of order 2 if rA,2-1(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of two not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z → N0∪ ∞ is any function such that f-1(0) is finite, then there exists a set A of integers such that f(n) = rA,2(n) for all n ∈ \Z. Moreover, the set A can be constructed so that carda∈ A : |a| ≤ x ≫ x1/3.