发表机构
Durham University（杜伦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究乘法函数中特定局部极值模式的频率，给出精确分类，并强形式解决Sárközy关于有限局部极值的两个猜想，证明基于加法函数的二分法及新间隙界。
AI 中文摘要
设 $f: \mathbb{N} \rightarrow (0,\infty)$ 为乘法函数，且 $\omega > 0$。我们考虑形如 $$ f(n) > \omega \max\{f(n-1),f(n+1)\}, \quad f(n) < \frac{1}{\omega}\min\{f(n-1),f(n+1)\} $$ 的模式在序列 $(f(n))_n$ 中出现的频率，并对所有正值乘法函数给出一个精确的分类，这些函数使得上述至少一种模式仅以对数密度 $0$ 出现。作为特别应用，我们以强形式解决了A. Sárközy在2001年提出的两个猜想，分类了所有具有有限多个局部最大值或有限多个局部最小值的乘法函数 $f: \mathbb{N} \rightarrow \mathbb{N}$。证明基于加法函数的自然“pretentiousness”二分法。分析中的关键工具包括作者提出的加法函数的Matomäki-Radziwiłł定理的类似物，以及关于加法函数 $g$ 的间隙 $g(n+1)-g(n)$ 集中度的新界，该界可能具有独立的意义。
英文摘要
Let $f: \mathbb{N} \rightarrow (0,\infty)$ be a multiplicative function, and let $ω> 0$. We consider the frequency with which patterns of the form $$ f(n) > ω\max\{f(n-1),f(n+1)\}, \quad f(n) < \frac{1}ω\min\{f(n-1),f(n+1)\} $$ occur in the sequence $(f(n))_n$, and give a sharp classification of all positive-valued multiplicative functions for which at least one of these patterns only occurs with logarithmic density $0$. As a particular application, we resolve, in a strong form, two conjectures of A. Sárközy from 2001, classifying all multiplicative functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that have either finitely many local maxima, or finitely many local minima. The proof is based on a natural ``pretentiousness'' dichotomy for additive functions. Key tools in the analysis include an analogue of the Matomäki-Radziwiłłtheorem for additive functions, due to the author, as well as a new bound for the concentration of the gaps $g(n+1)-g(n)$ of an additive function $g$, which may be of independent interest.
Comments27 pages
