Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

For any given positive integer $k$, and any set $A$ of nonnegative integers, let $r_{1, k}(A, n)$ denote the number of solutions of the equation $n=a_1+ka_2$ with $a_1, a_2\in A$. We prove that if $k,l$ are multiplicatively independent integers, i.e., $\log{k}/\log{l}$ is irrational, then there does not exist any set $A\subseteq \mathbb{N}$ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all $n\geq n_0$. We also pose a conjecture and two problems for further research.