Unique representations of integers by linear forms

Abstract

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda }=({\lambda }_1,\dots ,{\lambda }_k)$ with $1\le {\lambda }_1<{\lambda }_2<\dots <{\lambda }_k$, we define the additive representation function $R_{A,\underline{\lambda }}(n)=|\{(a_1,\dots ,a_k)\in A^k:{\lambda }_1{a}_1+\dots +{\lambda }_k{a}_k=n\}|$. For $k=2$, Moser constructed a set $A$ of nonnegative integers such that $R_{A,\underline{\lambda }}(n)=1$ holds for every nonnegative integer $n$. In this paper we characterize all the $k$-tuples $\underline{\lambda }$ and the sets $A$ of nonnegative integers with $R_{A,\underline{\lambda }}(n)=1$ for every integer $n\ge 0$.