Partitions Of Natural Numbers Which Has The Same Representation Functions

Let N denote the set of all nonnegative integers.For A(?)N and n ?N,let RA(n)denote the number of solutions of the equation a+a' =n,a,a'? A,a<a',Let {h1,h2,…}1<h2<…)be an infinite set of positive integers.DefineWe call H(hi,h2,...)a Hilbert-cube.A Hilbert-cube H is called a nondegenerate Hilbert-cube if ??ihi???'ihi for different {?i} and {?'i}.The study on partitions of natural numbers which has the same representation functions is an important subject in combinatorial number theory.In 2016,Chen and Lev posed three open problems in paper[Integer sets with identical represen-tation functions,Integers 16(2016),A36].In 2017,Kiss and Sandor solved one of problems.In this thesis,we focus on another open problem.The main results of this thesis are divided into three parts.In the first part,we study partitions of the set of nonnegative integers with the same representation functions and obtain the following result:Let m>r ? 0 be integers.Let A and B be sets of nonnegative integers such that A ?B = N and A?B= {r+mk:k?kN}.If RA(n)= RB(n)for every positive integer n,then there exists an integer l ?1 such that r= 22t-1.In the second part,we further characterize the properties of representation functions contructed by Chen and Lev.In the three part,we characterize the necessary and sufficient conditions of partition of a nondegenerate Hilbert-cube which has the same representation function and whose intersection.is nonempty,and we obtain the following result:Let H= H(h1,h2=,…)be a nondegenerate Hilbert-cube.Let C and D be two sets of nonnegative integers such that C?D = H,C ? D =(?)and 0 ? C.Then RC(n)=RDd(n)for every positive integer n if and only if C = H0(h1,h2,...),D ?H1(h1,h2,...).