First order of the renewal covering of the natural numbers

TL;DR

This paper studies a novel renewal process covering of natural numbers inspired by prime numbers, revealing a first-order limit behavior with a concentration around n log n, and highlighting its dependence on initial conditions.

Contribution

Introduces a new renewal covering process for natural numbers, analyzing its first-order asymptotic behavior and its similarities to prime number distribution.

Findings

The number of objects placed up to n concentrates around n log n.

Small initial perturbations lead to divergent outcomes.

The process exhibits prime-like distributional properties.

Abstract

This paper introduces a new type of covering process that covers the set of natural numbers using renewal processes as objects. Inspired by the behavior of prime numbers, the model in each step finds the smallest vacant point, $k$, and place, starting in $k$, a renewal process with a step distribution given by a geometric random variable with parameter $\frac{1}{k}$. The model depends on its entire past, and small perturbations in its initial value can lead to very different outcomes. Here, we expose a technique that finds the first-order limit behavior for the number of objects placed until $n$, which exhibits intriguing similarities to prime number distributions, having a concentration around $n\log{n}$.