On the asymptotic distribution of algebraic numbers with growing naive height

D. V. Koleda

Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 191-204

Résumé

Till recently, even for quadratic algebraic numbers, it was unknown, how frequently do algebraic numbers appear in an arbitrary interval depending on its position and length. Let $\mathbb{A}_n$ be the set of algebraic numbers of $n$-th degree, and let $H(\alpha)$ be the naive height of $\alpha$ that equals to the naive height of its minimal polynomial by definition. The above problem comes to the study of the following function: $$ \Phi_n(Q, x) := \# \left\{ \alpha \in \mathbb{A}_n \cap \mathbb{R} : H(\alpha)\le Q, \ \alpha x \right\}. $$ The exact asymptotics of $\Phi_n(Q,x)$ as $Q\to +\infty$ was recently obtained by the author. There, in fact, the density function of real algebraic numbers was correctly defined and explicitly described. In the paper, we discuss the results on the distribution of real algebraic numbers. For $n=2$, we improve an estimate of a remainder term in the asymptotics of $\Phi_2(Q,x)$, and obtain the following formula: $$ \Phi_2(Q, +\infty) = \lambda\, Q^3 - \kappa\, Q^2 \ln Q + O(Q^2), $$ where $\lambda$ and $\kappa$ are effective constants. Bibliography: 16 titles.

Détail

Keywords: algebraic numbers, generalized Farey series, integral polynomials.