Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond

TL;DR

This paper demonstrates how automata theory and Fibonacci representations can be used to analyze sumsets of Wythoff sequences, providing a unified and efficient method to derive known and new results.

Contribution

It introduces a simple automaton-based approach using Walnut to derive results about sumsets of Wythoff sequences and Fibonacci representations.

Findings

Automaton approach effectively derives sumset properties.

Automata accept Fibonacci representations of sumset elements.

Method simplifies proving known and new results.

Abstract

Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form $a_i + a_j$, $b_i + b_j$, $a_i + b_j$, and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented. I also show how the automaton approach can easily prove other results.