@article{Pongsriiam2022, 
author = {Prapanpong Pongsriiam},
title = {Combinatorial structure and sumsets associated with Beatty sequences generated by powers of the golden ratio},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {7},
pages = {2385-2405},
keywords = {golden ratio, Fibonacci number, Beatty sequence, sumset, Wythoff sequence, fractional part},
url = {https://www.sciopen.com/article/10.3934/era.2022121},
doi = {10.3934/era.2022121},
abstract = {Let  α be the golden ratio,  m∈N, and  B(αm) the Beatty sequence (or Beatty set) generated by  αm. In this article, we give some combinatorial structures of  B(αm) and use them in the study of associated sumsets. In particular, we obtain, for each  m∈N, a positive integer  h=h(m) such that the  h-fold sumset  hB(αm) is a cofinite subset of  N. In addition, we explicitly give the integer  N=N(m) such that  hB(αm) 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.}
}