Arithmetical independence of certain uniform sets of algebraic integers

TL;DR

This paper investigates four families of algebraic integers of degree up to three, demonstrating their almost uniform distribution and arithmetical independence, and explores how their generated number fields relate to quadratic and cubic fields.

Contribution

It introduces four specific sets of algebraic integers, analyzing their distribution and independence properties, and examines their coverage of quadratic and cubic number fields.

Findings

Elements are distributed almost equidistantly in the unit interval

Number fields generated by the sets are arithmetically independent

Extent of coverage of quadratic and cubic fields by these sets

Abstract

We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here, ``almost uniformity'' means that the elements of a finite set are distributed almost equidistantly in the unit interval, while ``arithmetical independence'' means that the number fields generated by the elements of a set do not have a mutual inclusion relation each other. Furthermore, we reveal to what extent the algebraic number fields generated by the elements of the four sets can cover quadratic or cubic fields.