The Parameterized Periodicity Lemma

2026-08-22

AI Summary

本文解决了参数化字符串的周期性问题，通过证明一个更优的Fine-Wilf类型界来确定当字符串长度满足一定条件时，其最大公约数也是周期。

Abstract

Fine and Wilf [Proc. Amer. Math. Soc. 1965] showed that any string of length at least $p+q-d$ with periods $p$ and $q$ also has period $d=\gcd(p,q)$. For parameterized strings, Apostolico and Giancarlo [Discrete Appl. Math. 2008] proved an analogue with length bound $p+q$, assuming that the two induced bijections commute. Ideguchi et al. [SPIRE 2023] removed this assumption and gave the bound $p+q+\min(p,q)(σ-1)$, where $σ$ is the number of distinct letters. This was later improved by Hamai et al. [SPIRE 2024] to $p+q+\min(p,q)(σ-2)$, which was used to bound the number of non-equivalent parameterized squares. In this paper, we establish the optimal Fine--Wilf type bound for parameterized strings. Namely, if a string $s$ containing $σ$ distinct letters has parameterized periods $p$ and $q$ and satisfies $|s| \ge p+q+(σ-3)d+1$, where $d=\gcd(p,q)$, then $d$ is also a parameterized period of $s$. We also give matching lower-bound instances, proving that our bound is optimal for any $σ\geq 2$.

Problem

• parameterized strings
• Fine-Wilf type bound
• periodicity
• gcd
• distinct letters

Innovation

• parameterized strings
• periodicity lemma
• optimal bound
• gcd