On a Problem of Q. H. YANG and Y. G. CHEN

作     者：Xiaohui YAN Xiaohui YAN

作者机构：School of Mathematics and Statistics Anhui Normal University 

出 版 物：《Journal of Mathematical Research with Applications》 (数学研究及应用(英文版))

年 卷 期：2022年第42卷第6期

页      面：580-586页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Supported by the National Natural Science Foundation of China (Grant No. 12101009) the Natural Science Foundation of Anhui Province (Grant No. 2108085QA02) 

主　　题：representation function partition Sarkozy problem 

摘      要：For any positive integers k1, k2and any set A■N, let Rk1,k2(A, n) be the number of solutions of the equation n = k1a1+ k2a2with a1, a2∈A. Let ■= NA. Yang and Chen proved that if k1and k2are two integers with k2k1≥2 and(k1, k2) = 1, then there does not exist any set A■N such that Rk1,k2(A, n) = Rk1,k2(■, n) for all sufficiently large integers n. For two integers k1 and t≥1, define fk(t) to be the number of sets A■N such that R1,k(A, n) = R1,k(■, n) holds for all integers n≥t. Yang and Chen proved that fk(t) is finite and limt→∞log fk(t)/t= log 2. They also asked if it is true that for any integers k, l 1 there exists t0(k, l) such that fk(t) = fl(t) for all integers t≥t0. In this paper, we give the exact formula of fk(t) when t≤k, which implies that fk(t) = fl(t) for all integers t≤min{k, l}.