Gap Sequence of Cutting Sequence with Slope $\theta=[0;\dot{d}]$

TL;DR

This paper analyzes the gap sequence of a specific cutting sequence with slope [0;dot{d}], revealing it has exactly two distinct gaps, and explores related combinatorial properties and palindrome structures.

Contribution

It introduces the kernel and envelope word framework to completely characterize the gap sequence and its properties for the sequence with slope [0;dot{d}].

Findings

The gap sequence has exactly two distinct elements for each factor.

The gap sequence can be explicitly expressed using a substitution based on kernel types.

Identifies all palindromes within the sequence.

Abstract

In this paper, we consider the factor properties and gap sequence of a special type of cutting sequence with slope $\theta=[0;\dot{d}]$, denoted by $F_{d,\infty}$. Let $\omega$ be a factor of $F_{d,\infty}$, 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 $\omega_{p+1}$. We define the $d$ types of kernel words and envelope words, give two versions of "uniqueness of kernel decomposition property". Using them, we prove the gap sequence $\{G_p(\omega)\}_{p\geq1}$ has exactly two distinct elements for each $\omega$, and determine the expressions of gaps completely. Furthermore, we prove that the gap sequence is $\sigma_i(F_{d,\infty})$, where $\sigma_i$ is a substitution depending only on the type of $Ker(\omega)$, i.e. the kernel word of $\omega$. We also determine the position of $\omega_p$ for all $(\omega,p)$. As applications, we study some combinatorial properties, such as the power, overlap and separate property between $\omega_p$ and $\omega_{p+1}$ for all $(\omega,p)$, and find all palindromes in $F_{d,\infty}$.