On S\'{a}rk\"{o}zy's Local Extrema Conjectures

Abstract

Let $f: \mathbb{N} \rightarrow (0,\infty)$ be a multiplicative function, and let $\omega>0$. We consider the frequency with which patterns of the form $$ f(n)>\omega \max\{f(n-1),f(n+1)\}, \quad f(n)<\frac{1}{\omega}\min\{f(n-1),f(n+1)\} $$ occur in the sequence $(f(n))_n$, and give necessary conditions on $f$ so that the set of occurrences of at least one of these patterns has logarithmic density $0$. This includes a complete classification of all such completely multiplicative functions. As a particular application, we characterise (up to prime powers $p^k$, $k \geq 2$) all multiplicative functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that have either finitely many local maxima, or finitely many local minima. This answers, in a strong form, two conjectures of A. S\'{a}rk\"{o}zy from 2001. \\ The proof is based on a natural ``pretentiousness''dichotomy for additive functions. Key tools in the analysis include an analogue of the Matom\"{a}ki-Radziwi\l\l theorem 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.