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Asymptotic $ψ$-densities of subsets of natural numbers

Janne Heittokangas, Zinelaabidine Latreuch

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.

Asymptotic $ψ$-densities of subsets of natural numbers

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 -densities, where a specific weight function plays a key role. In this paper, a parallel theory of asymptotic -densities is introduced, where the weight is expressed slightly differently in terms of differentiable functions , 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.
Paper Structure (11 sections, 23 theorems, 165 equations)

This paper contains 11 sections, 23 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Asymptotic $\psi$-densities
  3. Function classes $\mathcal{D}_1$ and $\mathcal{D}_2$
  4. Comparison of asymptotic $\psi$-densities
  5. Comparison of other densities
  6. Asymptotic $\psi$-density of strictly increasing sequences
  7. Regularity of asymptotic $\psi$-densities
  8. Regularity in $\mathcal{D}_1$ and $\mathcal{D}_2$
  9. More sufficient conditions for regularity in $\mathcal{D}_2$
  10. On the identity $\psi(n+1)\sim \psi(n)$
  11. Analytic and Abel densities

Key Result

Theorem 1.1

Let $(s_n)_n$ be a real sequence, and let $(a_n)_n$ and $(b_n)_n$ be positive real sequences such that Denote If $(a_n/b_n)_n$ is decreasing, then

Theorems & Definitions (47)

  • Theorem 1.1: R
  • Definition 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 37 more