Unique expansions in number systems via refinement equations

Using the subdivision scheme theory we develop a criterion to check if each natural number has at most one representation in the $n$-ary number system with a set of nonnegative integer digits $A=\{a_1, a_2,…, a_n\}$ that contains zero. This uniqueness property is shown to be equivalent to a certain restriction on the zeros 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 the case of a prime number $n$ 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 a connection of this uniqueness with the problem of semigroup freeness for affine integer functions of equal integer slope; in combination with the two criteria, this allows us to fill the gap in a work of Klarner on the question of Erdős about the densities of affine integer orbits and to establish a simple algorithm to check freeness and the positivity of density in the case when the slope is a prime number.

number system, semigroup freeness, refinement equation, subdivision scheme, Borel measure