Sumsets of Wythoff sequences, Fibonacci representation, and beyond

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 $$i \ge 1$$ . 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.