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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 31, 2015 12:30, 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:
$$ \Phi_n(Q, x) := # \{ \alpha \in \mathbb{A}_n \cap \mathbb{R} : H(\alpha)\le Q, \alpha < x \}. $$
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. фiз-мат. навук. — 2013. — № 3. — С. 54–63.

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