Runs in Paperfolding Sequences

TL;DR

This paper studies the run-length sequences of paperfolding sequences, showing they are 2-automatic and computable by finite automata, and explores their complexity and critical exponent.

Contribution

It demonstrates that run-lengths and positions in paperfolding sequences are 2-automatic, providing a new automata-based approach to their analysis.

Findings

Run-length sequences are 2-automatic and computable by finite automata.

Results on critical exponent and subword complexity of these sequences.

Generalizes recent results using automata theory.

Abstract

The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note we observe that the sequence of run lengths in such a sequence, as well as the starting and ending positions of the $n$'th run, is $2$-synchronized and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates, and Arnold, in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.