Envelope Word and Gap Sequence in Doubling Sequence

TL;DR

This paper analyzes the structure of gap sequences between repeated factors in the Doubling sequence, introducing the novel concept of envelope words to classify and determine gap patterns and positions.

Contribution

It introduces envelope words as a new tool to classify and analyze gap sequences in the Doubling sequence, providing explicit expressions and substitution rules.

Findings

Factors are classified into two types based on their gap patterns.

Explicit formulas for gap sequences are established.

Positions of all repeated factors and properties of squares are determined.

Abstract

Let $\omega$ be a factor of Doubling sequence $D_\infty=x_1x_2\cdots$, then it occurs in the sequence infinitely many times. Let $\omega_p$ be the $p$-th occurrence of $\omega$ and $G_p(\omega)$ be the gap between $\omega_p$ and $\omega_{p+1}$. In this paper, we discuss the structure of the gap sequence $\{G_p(\omega)\}_{p\geq1}$. We prove that all factors can be divided into two types, one type has exactly two distinct gaps $G_1(\omega)$ and $G_2(\omega)$, the other type has exactly three distinct gaps $G_1(\omega)$, $G_2(\omega)$ and $G_4(\omega)$. We determine the expressions of gaps completely. And also give the substitution of each gap sequence. The main tool in this paper is "envelope word", which is a new notion, denoted by $E_{m,i}$. As an application, we determine the positions of all $\omega_p$, discuss some combinatorial properties of factors, and count the distinct squares beginning in $D_\infty[1,N]$ for $N\geq1$.