Intrinsic Characterization of Representation Functions and Generalizations

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Given a set A of natural numbers, i.e., nonnegative integers, we establish an intrinsic characterization of the representation function of A, which to every natural number n associates the number rA(n) of ordered pairs (a, b) of elements a, b∈ A such that a+ b= n, thus answering an open problem stated many years ago by Mel Nathanson. We also establish similar characterizations of the characteristic function χA(n), which is equal to 1 or 0 according as the natural number n lies or does not lie in A, and the counting function A(n), which gives the number of elements a of A satisfying a≤ n. We then generalize to representation functions as sums of more than two elements of A.

Original languageEnglish (US)
Title of host publicationCombinatorial and Additive Number Theory IV, CANT 2019 and 2020
EditorsMelvyn B. Nathanson
PublisherSpringer
Pages291-305
Number of pages15
ISBN (Print)9783030679958
DOIs
StatePublished - 2021
EventWorkshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020 - Virtual, Online
Duration: Jun 1 2020Jun 5 2020

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume347
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceWorkshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020
CityVirtual, Online
Period6/1/206/5/20

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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