Distribution of real algebraic integers

Abstract

In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their na\"ıve height tends to infinity. For an arbitrary interval $I\subset \mathbb{R}$ and sufficiently large $Q>0$, we obtain an asymptotic formula for the number of algebraic integers $\alpha \in I$ of fixed degree $n$ and na\"ıve height $H(\alpha )\le Q$. In particular, we show that the real algebraic integers of degree $n$, with their height growing, tend to be distributed like the real algebraic numbers of degree $n-1$. However, we reveal two symmetric "plateaux", where the distribution of real algebraic integers statistically resembles the rational integers.

34 pages; introduction rewritten; main theorems reformulated; lemma 13 remade