(Don't) Mind the Gap

TL;DR

This work studies the complexity of binary substitution sequences, focusing on the Thue–Morse sequence generated by $0 \mapsto 01$ and $1 \mapsto 10$. It leverages the concept of right-special words with the core relation $p(n+1) = p(n) + s(n)$ to compute the full complexity $p_{\mathrm{TM}}(n)$, illustrating initial values and a path to a general formula. The paper then generalizes to non-constant-length (gapped) substitutions with integer expansion factors, proving central-patch growth and showing how gaps can raise complexity compared to constant-length cases; it also demonstrates a recoding technique via higher-block sliding codes to convert gapped substitutions into constant-length ones on a larger alphabet, enabling standard analysis and explicit complexity computations. Together, these results provide concrete methods to analyze and compare the complexity of substitution sequences, including gapped digit substitutions, with implications for symbolic dynamics and coding theory.

Abstract

Infinite sequences are of tremendous theoretical and practical importance, and in the Information Age sequences of 0s and 1s are of particular interest. Over the past century, the field of symbolic dynamics has developed to study sequences with entries from a finite set. In this paper we will show you an interesting method for constructing sequences, and then we'll show you a fun variant of the method. The we'll discuss what theoretical computer scientists and mathematicians call the "complexity" of a sequence. We conclude by showing a technique you can use to compute the complexity of a sequence, illustrating it on our concrete examples.

Figures (7)

• Figure 1: Right special tree for the Thue-Morse substitution from Cas97.
• Figure 2: The image of $(0, {\sf a})$, with letters depicted in red and green.
• Figure 3: Here are the first few levels of the substitution. From top to bottom: $({\sf a})$, $\mathcal{S}({\sf a})$, $\mathcal{S}^2({\sf a})$, and $\mathcal{S}^3({\sf a})$
• Figure 4: The adjacency structure in the substitution of a length 5 word.
• Figure 5: Complexities for digit substitutions with ${Q} = 3$ and digit sets $\mathcal{D} = \{-1, 0, 4\}$ (column 3) and $\mathcal{D}' = \{-1, 0, 1\}$ (column 4).

Theorems & Definitions (8)

• Definition 1
• Proposition 2: Cas97, Proposition 3.1
• Proposition 3
• Lemma 4: Central Patch Lemma
• proof
• Proposition 5
• proof
• Theorem 6