Sumsets associated with Wythoff sequences and Fibonacci numbers
This is an interesting contribution to some additive number theory questions arising from specific integer sequences associated to two specific Beatty sequences. The lower and upper Wythoff sequences (the name coming from a version of Nim studied by \textit{W. A. Wythoff} [Nieuw Archief (2) 7, 199--202 (1906,1907; JFM 37.0261.03)] \((a_n)\) and \((b_n)\) respectively are defined by \(a_n=\lfloor n\alpha\rfloor\) and \(b_n=\lfloor n\alpha^2\rfloor\) for \(n\ge 1\), where \(\alpha=(1+\sqrt{5})/2\) is the golden ratio. A number of results concerning integers that can be expressed in the form \(a_n+a_m\), \(a_n+b_m\), or \(b_n+b_m\) are given. In particular, the rather complicated patterns arising in the set of integers that can be written \(b_n+b_m\) are described, and there is an intriguing suggestion that (suitably renormalized) there is a self-similar or asymptotically fractal structure to the set. Recently \textit{J. Shallit} [https://arxiv.org/abs/2006.04177], motivated by this work, has shown how to construct a finite automaton for each of these sumsets that accepts the Fibonacci representation of the numbers with a representation as such a sum. This gives a different approach to the results here, along with other related results.
scientific article; zbMATH DE number 7364740
scientific article