Asymptotic ψ-densities of subsets of natural numbers

2024/03/12 by Janne Heittokangas, Heittokangas, Janne, Zinelaâbidine Latreuch +1
Mathematics · #11B05 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2403.07600

openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The sizes of subsets of the natural numbers are typically quantified in terms of asymptotic (linear) and logarithmic densities. These concepts have been generalized to weighted w-densities, where a specific weight function w plays a key role. In this paper, a parallel theory of asymptotic ψ-densities is introduced, where the weight is expressed slightly differently in terms of differentiable functions ψ, which are either concave or convex and satisfy certain asymptotic properties. Alternative new proofs for known results on analytic and Abel densities are also given.

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