Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals
<|reference_start|>Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals: We show that the $n$'th digit of the base-$b$ representation of the golden ratio is a finite-state function of the Zeckendorf representation of $b^n$, and hence can be computed by a finite automaton. Similar results can be proven for any quadratic irrational. We use a satisfiability (SAT) solver to prove, in some cases, that the automata we construct are minimal.<|reference_end|>
@article{barnoff2024using,
    title={Using finite automata to compute the base-$b$ representation of the
  golden ratio and other quadratic irrationals},
    author={Aaron Barnoff, Curtis Bright, Jeffrey Shallit},
    journal={arXiv preprint arXiv:2405.02727},
    year={2024},
    doi={10.1007/978-3-031-71112-1_3},
    archivePrefix={arXiv},
    eprint={2405.02727},
    primaryClass={cs.FL cs.DM math.NT}
}