Distribution of real algebraic integers

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.

21 pages; Theorem 3 and comments added; typos corrected

References in corpus (1)

• On the density function of the distribution of real algebraic numbers