参数化周期性引理
The Parameterized Periodicity Lemma
AI总结：
本文针对参数化字符串的Fine-Wilf型周期界问题，突破以往研究的假设限制与界值精度，推导得出最优参数化周期界公式，并通过匹配下界实例验证其对任意σ≥2的最优性，完善了参数化周期性理论。
AI中文摘要：
Fine和Wilf于1965年在《美国数学会会刊》上证明，任何长度至少为$p+q-d$、具有周期$p$和$q$的字符串，也具有周期$d=\text{gcd}(p,q)$。对于参数化字符串，Apostolico和Giancarlo于2008年在《离散应用数学》中证明了一个类似结论，在假设两个诱导双射可交换的前提下，长度界为$p+q$。Ideguchi等人在2023年SPIRE会议上移除了该假设，给出的界为$p+q+\text{min}(p,q)(σ-1)$，其中$σ$是不同字母的数量。Hamai等人在2024年SPIRE会议上将其改进为$p+q+\text{min}(p,q)(σ-2)$，该结果被用于界定非等价参数化平方的数量。本文建立了参数化字符串的最优Fine-Wilf型界。具体而言，若包含$σ$个不同字母的字符串$s$具有参数化周期$p$和$q$，且满足$|s| \text{ge} p+q+(σ-3)d+1$（其中$d=\text{gcd}(p,q)$），则$d$也是$s$的参数化周期。本文还给出了匹配的下界实例，证明该界对于任意$σ\text{geq}2$都是最优的。
英文摘要：
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 $σ\geq 2$.
