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 $n\ge 0$.