

Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 31, 2015 12:30–12:55, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences






On the asymptotic distribution of algebraic numbers on the real axis
D. V. Koleda^{} ^{} Institute of Mathematics of the National Academy of Sciences of Belarus

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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 in [1],[2].
There, in fact, the density function of real algebraic numbers was correctly defined and explicitly described.
In the talk, we will discuss the results [1],[2] on the distribution of real algebraic numbers.
[1] Каляда Д.У. Аб размеркаваннi рэчаiсных алгебраiчных лiкаў дадзенай ступенi. — Доклады НАН Беларуси. — 2012. — Т. 56, № 3. — С. 28–33.
[2] Коледа Д.В. О распределении действительных алгебраических чисел второй степени. — Весцi НАН Беларусi. Сер. фiзмат. навук. — 2013. — № 3. — С. 54–63.

