Completeness of exponentially increasing sequences

For fixed positive reals $t$ and $\alpha$, consider the sequence $S_t(\alpha) = (s_1, s_2, \ldots, )$ with $s_n = \left \lfloor t\alpha^n \right \rfloor$. In 1964, Graham managed to characterize those pairs $(t, \alpha)$ with $0 < t < 1$ and $1 < \alpha < 2$ for which every large enough integer can be written as the sum of distinct elements of $S_t(\alpha)$. We show that his methods can be applied to deal with many other pairs of $(t, \alpha)$ as well.