Sumsets associated with Wythoff sequences and Fibonacci numbers
type: Item
instance of: scholarly article
title: Sumsets associated with Wythoff sequences and Fibonacci numbers
zbMATH Open document ID: 1474.11033
author: Sutasinee Kawsumarng
author: Tammatada Khemaratchatakumthorn
author: Passawan Noppakaew
author: Prapanpong Pongsriiam
published in: Periodica Mathematica Hungarica
publication date: 2021-06-29T00:00:00Z
review text: 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.
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 7364740
zbMATH Keywords: Wythoff sequence
zbMATH Keywords: sumset
zbMATH Keywords: Fibonacci number
zbMATH Keywords: Golden ratio
zbMATH Keywords: Beatty sequence
MaRDI profile type: Publication
full work available at URL: s10998-020-00343-0
OpenAlex ID: W3024801220
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DOI: 10.1007/S10998-020-00343-0
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description: scientific article; zbMATH DE number 7364740