Distribution of real algebraic integers (Q2291712)

In a previous paper [J. Théor. Nombres Bordx. 29, No. 1, 179--200 (2017; Zbl 1420.11126)], the author studied the asymptotic distribution of the real algebraic numbers of fixed degree as their naïve height tends to infinity. In the paper under review, he studies the same question for real algebraic integers, using an ordering inspired by the generalized Farey sequence [\textit{H. Brown} and \textit{K. Mahler}, J. Number Theory 3, 364--370 (1971; Zbl 0221.10014)]. For \(n\ge 2\), \(Q>1\) and \(I\) an interval of the real line, he introduces the counting function \(\Omega_n(Q,I)\) as the number of algebraic integers in the interval \(I\) having degree \(n\) and naive height at most \(Q\). He shows that the limit as \(Q\to\infty\) of \(\Omega_n(Q,I)/(2Q)^n\) exists and he gives an explicit formula for the distribution density. He shows that it is essentially the same as the distribution density of the real algebraic numbers of degree \(n-1\), with the exception of two symmetric plateaux, for \(I \subset (-Q - 1, -Q^{1/2}) \cup (Q^{1/2}, Q + 1)\), due mainly to Perron numbers. The author also provides references to counting functions with respect to height functions other than the naïve height.