Given a natural number $n>1$. It is well-known that any real number $r$ can be written as $$\sum_{i=-\infty}^{\infty}a_in^i$$ such that $a_i\in \left\{0,1,\dots n-1\right\}=C$.
For the decimal system $n=10$ and in the binary system $n=2$.
I was wondering whether a similar statement holds when $n\in \mathbb{Q}$ (or even $\mathbb{R}$). For example if $n=\frac{3}{2}$, how many coefficients do we need to allow for this to work? If $C=\mathbb{Z}$ it is still possible. But what is the smallest coefficient set we need to allow?
I'm fairly certain someone has thought about this and something is known. I'd be very happy with a reference on this subject.
As stated in the comments, this question has a positive answer:
Let $b\in\mathbb{R}$ and assume $b>1$. Then any real number $r$ can be written as $$\sum_{i=-\infty}^{\infty}a_ib^i$$ where $a_i\in \left\{0,1,\dots ,n\right\}$ where $n$ is the largest natural number smaller than $n$. One can verify that leaving out a coefficient gives rise to certain real numbers that can no longer be expressed in this way.
References can be found in Ethan Bolker's comment and here.