The author addresses the question of the behavior of continued fraction algorithms in terms of acceptance by finite automata. More precisely, call \(f: \mathbb R\to\mathbb Z\) a “real integer function” if \(\forall x\in\mathbb R\), \(|f(x)- x|<1\) and \(\forall n\in\mathbb R\), \(\forall j\in\mathbb Z\), \(f(x+j)= f(x)+ j\) (examples are the floor and the ceiling functions, and the nearest integer function). With such a function one can associate a continued fraction algorithm \(Af\) exactly the same way as in the classical case (that corresponds to \(f(x)= \lfloor x\rfloor\)).
The main theorem of this paper is that if \(f^{-1}(0)\) is a finite union of intervals, then the set of outputs of \(Af\) is recognized by a finite automaton if and only if all the endpoints of these intervals are rational or quadratic. In particular, the result holds for the usual, the ceiling, and the nearest integer continued fraction algorithms, as well as for the general Tanaka-Itô continued fraction algorithm [S. Tanaka and S. Itô, Tokyo J. Math. 4, 153–175 (1981; Zbl 0464.10038)] that corresponds to \(f(x)= \lfloor x-\alpha+ 1\rfloor\) with \(\alpha\in [{1\over 2},1]\).
For the entire collection see [Zbl 1266.11001].