Distribution of real algebraic integers

TL;DR

This paper investigates how real algebraic integers of fixed degree are distributed within bounded intervals as their naive height grows large, providing asymptotic formulas and error estimates.

Contribution

It presents the first asymptotic formula for counting real algebraic integers of fixed degree within an interval, including error bounds, and compares their distribution to algebraic numbers of lower degree.

Findings

Algebraic integers of degree n are asymptotically distributed like algebraic numbers of degree n-1.

An asymptotic formula for counting algebraic integers with bounded naive height is derived.

Error terms in the distribution estimate are explicitly bounded from above and below.

Abstract

In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naive height tends to infinity. Let $I \subset \mathbb{R}$ be an arbitrary bounded interval, and $Q$ be a sufficiently large number. We obtain an asymptotic formula for the count of algebraic integers $\alpha$ of fixed degree $n$ and naive height $H(\alpha)\le Q$ lying in $I$. In this formula, we estimate the order of the error term from above and below. We show that algebraic integers of degree $n$ are distributed asymptotically like algebraic numbers of degree $(n-1)$ as the upper bound $Q$ of heights tends to infinity.