Abstract:
Using 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 $A=\{a_1,a_2,\dots, a_n\}$ 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 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 a gap in the work of Klarner on the question of Erdős about the 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.
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