Completeness of exponentially increasing sequences

Abstract

For fixed positive reals $t$ and $\alpha$, consider the sequence $S_t(\alpha )=(s_1,s_2,\dots ,)$ with $s_n=\lfloor t{\alpha }^n\rfloor$. 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.

arXiv comment11 pages