On Sárközy's Local Extrema Conjectures
Abstract
Let f: ℕ → (0,∞) be a multiplicative function, and let ω> 0. We consider the frequency with which patterns of the form f(n) > ωmax{f(n−1),f(n+1)}, f(n) < 1/ωmin{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 pk, k ≥ 2) all multiplicative functions f: ℕ → 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.
Related
- Local Extrema and Infinite Derivatives of the Delange Function of Base 3
- An explicit power-saving bound for Sárközy's Theorem on shifted primes, assuming GRH
- Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture
- A negative answer to the Erdős-Sárkőzy question
- A Generalization of Sárközy's theorem in function fields
BibTeX
@misc{mangerel2026sárközys,
title = {{On Sárközy's Local Extrema Conjectures}},
author = {Alexander P. Mangerel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08424},
eprint = {2610.08424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08424},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Sárközy's Local Extrema Conjectures
AU - Alexander P. Mangerel
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.08424
UR - https://arxiv.org/abs/2610.08424
AB - Let f: ℕ → (0,∞) be a multiplicative function, and let ω> 0. We consider the frequency with which patterns of the form f(n) > ωmax{f(n−1),f(n+1)}, f(n) < 1/ωmin{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 ≥ 2) all multiplicative functions f: ℕ → 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.
ER -