TY - GEN

T1 - Intrinsic Characterization of Representation Functions and Generalizations

AU - Helou, Charles

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-67996-5_15

DO - 10.1007/978-3-030-67996-5_15

M3 - Conference contribution

AN - SCOPUS:85115184322

SN - 9783030679958

T3 - Springer Proceedings in Mathematics and Statistics

SP - 291

EP - 305

BT - Combinatorial and Additive Number Theory IV, CANT 2019 and 2020

A2 - Nathanson, Melvyn B.

PB - Springer

T2 - Workshops on Combinatorial and Additive Number Theory, CANT 2019 and 2020

Y2 - 1 June 2020 through 5 June 2020

ER -