Using finite automata to compute the base-$b$ representation of the golden ratio and other quadratic irrationals

TL;DR

This paper demonstrates that the base-$b$ representation of the golden ratio and other quadratic irrationals can be computed using finite automata, linking digit extraction to Zeckendorf representations and automaton minimality verified by SAT solvers.

Contribution

It introduces a novel method to compute quadratic irrational base-$b$ expansions via finite automata based on Zeckendorf representations, with minimality proofs using SAT solvers.

Findings

Digits of the golden ratio are finite-state functions of Zeckendorf representations.

Automata for quadratic irrationals can be constructed and verified for minimality.

The approach generalizes to other quadratic irrationals beyond the golden ratio.

Abstract

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.