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