Arithmetic Independence of Certain Uniform Sets of Algebraic Integers

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 arithmetic independence. Here, “almost uniformity” means that the elements of a finite set are distributed lmost equidistantly in the unit interval, while “arithmetic 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.