In this paper by erdos, at the beginning of the proof of theorem $ 1$ he considers the case where both $ \log p$ and $ \log q$ are commensurable numbers(which means that $ p=r^{l}$ and $ q=r^{m}$ ) Further states that any sum $ \sum_{i}r^{n_{i}kl}$ has all the digits either $ 0$ or $ […]