On Sárközy's Local Extrema Conjectures

Abstract

Let $f:\mathbb{N}\to (0,\infty )$ be a multiplicative function, and let $\omega >0$. We consider the frequency with which patterns of the form $$f(n)>\omega \max \{f(n-1),f(n+1)\},f(n)<\frac{1}{\omega }\min \{f(n-1),f(n+1)\}$$ 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 $p^k$, $k\ge 2$) all multiplicative functions $f:\mathbb{N}\to \mathbb{N}$ that have either finitely many local maxima, or finitely many local minima. This answers, in a strong form, two conjectures of A. Sárközy from 2001. \\ 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.

arXiv comment39 pages; fixed minor gaps, added appendix