Using the subdivision schemes theory, we develop a criterion to check if any natural number has at most one representation in the $n$-ary number system with a set of non-negative integer digits $A=\{a_1, a_2,\ldots, a_n\}$ that contains zero. This uniqueness property is shown to be equivalent to a certain restriction on the roots of the trigonometric polynomial $\sum_{k=1}^n e^{-2\pi i a_k t}$. From this criterion, under a natural condition of irreducibility for $A$, we deduce that in case of prime $n$ the uniqueness holds if and only if the digits of $A$ are distinct modulo $n$, whereas for any composite $n$ we show that the latter condition is not necessary. We also establish the connection of this uniqueness to the semigroup freeness problem for affine integer functions of equal integer slope; this together with the two criteria allows to fill the gap in the work of D. Klarner on the question of P. Erd\"os about densities of affine integer orbits and establish a simple algorithm to check the freeness and the positivity of density when the slope is a prime number.

翻译：利用细分格式理论，我们建立了一个判别准则，用于检验任意自然数在包含零的非负整数数字集$A=\{a_1, a_2,\ldots, a_n\}$构成的$n$进制数制中是否至多存在一种表示。该唯一性被证明等价于三角多项式$\sum_{k=1}^n e^{-2\pi i a_k t}$的根需满足特定约束。基于此准则，在$A$满足不可约的自然条件下，我们推导出：当$n$为素数时，唯一性成立当且仅当$A$中数字模$n$两两不同；而对于任意合数$n$，我们证明后一条件并非必要。此外，我们将此唯一性与等整数斜率的仿射整数函数半群自由性问题建立联系；结合两个判别准则，这填补了D. Klarner关于P. Erdős仿射整数轨道密度问题研究中的空白，并建立了一个简单算法来检验斜率为素数时的自由性及密度正性。