Representation functions with different weights

Quan-Hui Yang¹

¹ School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing 210044, China

Colloquium Mathematicum, Tome 137 (2014) no. 1, pp. 1-6.

Résumé

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.

Détail

DOI : 10.4064/cm137-1-1

Keywords: given positive integer set nonnegative integers denote number solutions equation prove multiplicatively independent integers log log irrational there does exist set subseteq mathbb mathbb setminus mathbb setminus geq pose conjecture problems further research