
This article is cited in 4 scientific papers (total in 5 papers)
INTERNATIONAL CONFERENCE IN MEMORY OF A. A. KARATSUBA ON NUMBER THEORY AND APPLICATIONS
On the asymptotic distribution of algebraic numbers with growing naive height
D. V. Koleda^{} ^{} Institute of Mathematics of the National Academy of Sciences of Belarus
Abstract:
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.
Keywords:
algebraic numbers, generalized Farey series, integral polynomials.
Received: 04.02.2015
Citation:
D. V. Koleda, “On the asymptotic distribution of algebraic numbers with growing naive height”, Chebyshevskii Sb., 16:1 (2015), 191–204
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https://www.mathnet.ru/eng/cheb375 https://www.mathnet.ru/eng/cheb/v16/i1/p191

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