Sets With Almost Coinciding Representation Functions

Abstract

For a given integer n and a set S ⊆ N, denote by R (1) h,S (n) the number of solutions of the equation n = s i1 + · · · + s ih , s i j ∈ S, j = 1, . . . , h. In this paper we determine all pairs (A, B), A, B ⊆ N, for which R (1) 3,A (n) = R (1) 3,B (n) from a certain point on. We discuss some related problems.2010 Mathematics subject classification: primary 11B34.

An extension of Nathanson’s Theorem on representation functions

“…In [4] they proved the sufficiency part of the Conjecture, and they also proved the Conjecture for the case h = 3.…”

“…Kiss, R. Rozgonyi and Cs. Sándor [4] conjectured that Nathanson's theorem can be generalized as follows.…”

Partitions of the Set of Nonnegative Integers With the Same Representation Functions

“…In [4], Kiss and Sándor give further conjectures and extensions of these results and partially describe the structure of the sets which have coinciding representation functions.…”

On a Partition Problem of Finite Abelian Groups

“…Nathanson [4] determined all pairs of sets A, B ⊆ N such that r A and r B eventually coincide. Kiss et al [2] extended Nathanson's result to 3-fold representation functions. In [3,6,8], the authors classified all subsets A ⊆ N such that r + A and r + N\A (respectively r − A and r − N\A ) eventually coincide.…”