Unique expansions in number systems via refinement equations

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.