Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond

DOI10.1007/S10998-021-00390-1arXiv2006.04177MaRDI QIDQ6342289

Publication date: 7 June 2020

Abstract: Let

a l p h a = (1 + s q r t 5) / 2

and define the lower and upper Wythoff sequences by

a_{i} = l f l o o r i a l p h a f l o o r

,

b_{i} = l f l o o r i a l p h a^{2} f l o o r

for

i g e q 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.

Mathematics Subject Classification ID

Formal languages and automata (68Q45) Additive bases, including sumsets (11B13) Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.) (68V15)

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