Combinatorial structure and sumsets associated with Beatty sequences generated by powers of the golden ratio

Let $\alpha$ be the golden ratio, $m\in \mathbb{N}$, and $B({\alpha }^m)$ the Beatty sequence (or Beatty set) generated by ${\alpha }^m$. In this article, we give some combinatorial structures of $B({\alpha }^m)$ and use them in the study of associated sumsets. In particular, we obtain, for each $m\in \mathbb{N}$, a positive integer $h=h(m)$ such that the $h$-fold sumset $hB({\alpha }^m)$ is a cofinite subset of $\mathbb{N}$. In addition, we explicitly give the integer $N=N(m)$ such that $hB({\alpha }^m)$ contains every integer that is larger than or equal to $N$, and show that this choice of $N$ is best possible when $m$ is small. We also propose some possible research problems. This paper extends the previous results on sumsets associated with upper and lower Wythoff sequences.