Further Research On Representation Functions

Let N denote the set of all nonnegative integers.For A CN and n G N,let R1{A,n)5 R2{A,n),R3{A,n)denote the number of solutions of a+a'=n,a,a'?A,a+a'=n,a,a'?A,a<a',a+a'=n,a,a'?A,a ?a'.respectively.The above three functions are called additive representation functions.For convenience,we write R2(A,n)=RA(n).Partitions of natural number sets and their corresponding additive represen-tation functions is an important subject in the field of additive representation functions.In this paper,we partially solve a problem posed in paper[Integer sets with identical representation functions,Integers 16(2016),A36].We also study an open question in[Partitions of the set of natural numbers and their representation functions,Discrete Math.308(2008),2614-2616].The main results of this thesis are divided into two parts.In the first part,we obtain the following result:Let m>r>0 be integers.Let A and B be sets such that A ? B=N and A ? B={r+mk.:k?N}.If RA(n)=RB(n)for every positive integer n,then there exists an integer l? 1 such that r=22l-1 and m=22l+1-1.In second part,we consider the problem of multidimensional partition,and obtain the following result:Given a positive integer k? 3,there is no partition such that RAu(n)=RAv(n)for every nonnegative integer n.Moreover,we also obtain the following result:Let k? 2 be a positive integer.There exists a partition N=UAi,Au?Av=?,1?u?v?k such that RAi(n)=RAk+1-i(n)(i=1,...,k)for every nonnegative integer n.