Completeness of exponentially increasing sequences

TL;DR

This paper extends Graham's 1964 characterization of when exponential sequences with fixed parameters can represent all large integers as sums of distinct elements, broadening the understanding of their completeness.

Contribution

It generalizes Graham's results to a wider range of parameters, demonstrating the applicability of his methods to more exponential sequences.

Findings

Graham's characterization applies to a broader set of parameters.

Methods extend to many other pairs of (t, alpha).

Sequences can represent all large integers as sums of distinct elements.

Abstract

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.