Complexity of natural numbers and arithmetic compact sets

TL;DR

This paper investigates the complexity of natural numbers based on minimal symbolic representations, introduces the concept of stable complexity, and explores the self-similar, well-ordered structure of certain fractional sets derived from these complexities.

Contribution

It introduces the notion of stable complexity for natural numbers and analyzes the self-similar, well-ordered properties of associated fractional sets.

Findings

The set of fractions derived from complexities exhibits self-similarity.

The fractional set is well-ordered.

The set has arithmetical properties and a remarkable structure.

Abstract

The complexity $\Vert n\Vert$ of a natural number is the least number of $1$ needed to represent $n$ using the 5 symbols $(, ), *, +, 1$. A natural number $n$ is called stable is $\Vert 3^kn\Vert =\Vert n\Vert +3k$. For each natural number $n$, the number $3^an$ is stable for some $a\ge0$, and we define the stable complexity of $n$ as $\Vert n \Vert _{\rm st}=\Vert 3^an\Vert -3a$. We show that the closure of the set of all fractions $n/3^{\lfloor \Vert {n}\Vert _{\rm st}/3\rfloor}$ has remarkable properties; self-similarity $3K'''=K$, well-ordered, and certain arithmetical properties. We pose the question about the unicity of this compact. This question raises some problems about the complexity of natural numbers that we are unable to answer.