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

Abstract

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.

Keywords

• beatty sequence
• sumset
• wythoff sequence
• fibonacci number
• golden ratio
• fractional part