A note on representation functions with different weights

Zhenhua Qu¹

¹ Department of Mathematics Shanghai Key Laboratory of PMMP East China Normal University 500 Dongchuan Rd. Shanghai 200241, China

Colloquium Mathematicum, Tome 143 (2016) no. 1, pp. 105-112.

Résumé

For any positive integer $k$ and any set $A$ of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions $(a_1,a_2)$ of the equation $n=a_1+ka_2$ with $a_1,a_2\in A$. Let $k,l\geq 2$ be two distinct integers. We prove that there exists a 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 $n\geq n_0$ if and only if $\log k/\!\log l=a/b$ for some odd positive integers $a,b$, disproving a conjecture of Yang. We also show that for any set $A\subseteq\mathbb N$ satisfying $r_{1,k}(A,n)=r_{1,k}(\mathbb N\setminus A,n)$ for all $n\geq n_0$, we have $r_{1,k}(A,n)\rightarrow\infty$ as $n\to\infty$.

Détail

DOI : 10.4064/cm6512-12-2015

Keywords: positive integer set nonnegative integers denote number solutions equation geq distinct integers prove there exists set subseteq mathbb mathbb setminus mathbb setminus geq only log log odd positive integers disproving conjecture yang set subseteq mathbb satisfying mathbb setminus geq have rightarrow infty infty