On sets with sum and difference structure

2022/05/13 by Fang, Jin-Hui, Sándor, Csaba
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.06553

Abstract

For nonempty sets A,B of nonnegative integers and an integer n, let r_A,B(n) be the number of representations of n as a+b and d_A,B(n) be the number of representations of n as a-b, where a∈ A, b∈ B. In this paper, we determine the sets A,B such that r_A,B(n)=1 for every nonnegative integer n. We also consider the difference structure and prove that: there exist sets A and B of nonnegative integers such that r_A,B(n)≥ 1 for all large n, A(x)B(x)=(1+o(1))x and for any given nonnegative integer c, we have d_A,B(n)=c for infinitely many positive integers n. Other related results are also contained.

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