ON THE ASYMPTOTIC DISTRIBUTION OF ALGEBRAIC NUMBERS WITH GROWING NAIVE HEIGHT
https://doi.org/10.22405/2226-8383-2015-16-1-191-204
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 An be the set of algebraic numbers of n-th degree, and let H(α) be the naive height of α that equals to the naive height of its minimal polynomial by definition. The above problem comes to the study of the following function: Φn(Q, x) := # {α ∈ An ∩ R : H(α) 6 Q, α < x} . The exact asymptotics of Φn(Q, x) as Q → +∞ 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 Φ2(Q, x), and obtain the following formula: Φ2(Q, +∞) = λ Q3 − κ Q2 ln Q + O(Q 2 ), where λ and κ are effective constants.
References
1. Baker, A. & Schmidt, W. 1970, “Diophantine approximation and Hausdorff dimension”, Proc. London Math. Soc., vol. 21, no. 3, pp. 1–11.
2. Bernik, V. I. 1989, “The exact order of approximating zero by values of integral polynomials”, Acta Arith., vol. 53, no. 1, pp. 17–28. (In Russian)
3. Beresnevich, V. 1999, “On approximation of real numbers by real algebraic numbers”, Acta Arith., vol. 90, no. 2, pp. 97–112.
4. Bernik, V. I. & Vasil’ev, D. V. 1999, “A Khinchin-type theorem for integervalued polynomials of a complex variable”, Trudy Instituta Matematiki, Natl. Akad. Nauk Belarusi, Inst. Mat., vol. 3, pp. 10–20. (In Russian) 5. Koleda, D. V. 2012, “Distribution of real algebraic numbers of a given degree”, Dokl. Nats. Akad. Nauk Belarusi, vol. 56, no. 3, pp. 28–33. (In Belarusian)
5. Koleda, D. V. 2013, “On the number of polynomials with a given number of roots on a finite interval”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, no. 1, pp. 41–49. (In Russian)
6. Koleda, D. V. 2013, “Distribution of real algebraic numbers of the second degree”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, no. 3, pp. 54–63. (In Russian)
7. Masser, D. & Vaaler, J. D. 2008, “Counting Algebraic Numbers with Large Height I”, Diophantine Approximation. Developments in Mathematics, vol. 16, pp. 237–243.
8. van der Waerden, B. L. 1936, “Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt”, Monatshefte f¨ur Mathematik, vol. 43, no. 1, pp. 133– 147.
9. Prasolov, V. V. 2001, Mnogochleny [Polynomials], 2-nd ed., MCCME, Moskow. (Russian)
10. Davenport, H. 1951, “On a principle of Lipschitz”, J. London Math. Soc., vol. 26, pp. 179–183. Davenport, H. 1964, “Corrigendum: «On a principle of Lipschitz»”, J. London Math. Soc., vol. 39, pp. 580.
11. Dubickas, A. 2014, “On the number of reducible polynomials of bounded naive height”, Manuscripta Mathematica, vol. 144, no. 3–4, pp. 439–456.
12. Mikol´as, M. 1949, “Farey series and their connection with the prime number problem. I”, Acta Univ. Szeged. Sect. Sci. Math., vol. 13, pp. 93–117.
13. Niederreiter, H. 1973, “The distribution of Farey points”, Math. Ann., vol. 201, pp. 341–345.
14. Brown, H. & Mahler, K. 1971, “A generalization of Farey sequences: Some exploration via the computer”, J. Number Theory, vol. 3, no. 3, pp. 364–370.
15. Cobeli, C. & Zaharescu, A. 2003, “The Haros-Farey sequence at two hundred years”, Acta Univ. Apulensis Math. Inform., no. 5, pp. 1–38.
Review
For citations:
Koleda D.V. ON THE ASYMPTOTIC DISTRIBUTION OF ALGEBRAIC NUMBERS WITH GROWING NAIVE HEIGHT. Chebyshevskii Sbornik. 2015;16(1):191-204. (In Russ.) https://doi.org/10.22405/2226-8383-2015-16-1-191-204
Views:
420
Email this article 