Question in a paper by Erdős on divisibility properties of central binomial coefficient

Question

In Erdős, Graham, Ruzsa, and Straus - On the prime factors of $\binom{2n}n$, at the beginning of the proof of theorem 1, they consider the case where $\log p$ and $\log q$ are commensurable numbers (which means that $p=r^{l}$ and $q=r^{m}$)

Further they state that any sum $\sum_{i}r^{n_{i}kl}$ has all the digits either $0$ or $1$ to both the bases $p$ and $q$. I didn't get this line particularly. Why are they taking $0$ and $1$ as the digits and what is $n_{i}$?

Answer 1

The $n_i$ are any sequence of distinct integers, indexed by $i$. The point is that if $R = r^{kl}$ then $R$ is a power of $p$, so any sum of distinct powers of $R$ is a 0-1 number in base $p$; and symmetrically the same is true of $q$.