On the density function of the distribution of real algebraic numbers

Denis Koleda

doi:10.5802/jtnb.975

Denis Koleda ¹

¹ Institute of Mathematics National Academy of Sciences of Belarus, Surganov street 11, 220072 Minsk, Belarus

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 179-200.

Résumé
Abstract

Dans cet article, nous étudions la distribution des nombres algébriques réels. Étant donné un intervalle $I$ , un entier positif $n$ et $Q > 1$ , on définit la fonction $Φ_{n} (Q; I)$ comme étant le nombre de nombres algébriques dans $I$ de degré $n$ et hauteur naïve $\leq Q$ . Soit $I_{x} = (- \infty, x]$ . La fonction de distribution est définie comme la limite (quand $Q \to \infty$ ) de $Φ_{n} (Q; I_{x})$ divisé par le nombre total de nombres algébriques réels de degré $n$ et de hauteur naïve $\leq Q$ . Nous montrons que la fonction de distribution existe et est continûment différentiable. Nous donnons aussi une formule explicite pour sa dérivée (dénommée la densité de la distribution). Nous établissons une formule asymptotique pour $Φ_{n} (Q; I)$ avec des estimations supérieure et inférieure pour le terme d’erreur dans cette formule. Il est démontré que ces estimations sont exactes pour $n \geq 3$ . Une conséquence du théorème principal est le fait que la distribution des nombres réels algébriques de degré $n \geq 2$ est non uniforme.

In this paper we study the distribution of the real algebraic numbers. Given an interval $I$ , a positive integer $n$ and $Q > 1$ , define the counting function $Φ_{n} (Q; I)$ to be the number of algebraic numbers in $I$ of degree $n$ and height $\leq Q$ . Let $I_{x} = (- \infty, x]$ . The distribution function is defined to be the limit (as $Q \to \infty$ ) of $Φ_{n} (Q; I_{x})$ divided by the total number of real algebraic numbers of degree $n$ and height $\leq Q$ . We prove that the distribution function exists and is continuously differentiable. We also give an explicit formula for its derivative (to be referred to as the distribution density) and establish an asymptotic formula for $Φ_{n} (Q; I)$ with upper and lower estimates for the error term in the asymptotic. These estimates are shown to be exact for $n \geq 3$ . One consequence of the main theorem is the fact that the distribution of real algebraic numbers of degree $n \geq 2$ is non-uniform.

Reçu le : 2015-03-30
Révisé le : 2015-07-12
Accepté le : 2015-07-20
Publié le : 2017-01-26

DOI : 10.5802/jtnb.975

Classification : 11N45, 11J83, 11K38
Mots clés : real algebraic numbers, distribution of algebraic numbers, integral polynomials, generalized Farey sequences

Affiliations des auteurs :

Denis Koleda ¹

¹ Institute of Mathematics National Academy of Sciences of Belarus, Surganov street 11, 220072 Minsk, Belarus

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2017__29_1_179_0,
     author = {Denis Koleda},
     title = {On the density function of the distribution of real algebraic numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {179--200},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {1},
     year = {2017},
     doi = {10.5802/jtnb.975},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.975/}
}

TY  - JOUR
AU  - Denis Koleda
TI  - On the density function of the distribution of real algebraic numbers
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2017
SP  - 179
EP  - 200
VL  - 29
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.975/
DO  - 10.5802/jtnb.975
LA  - en
ID  - JTNB_2017__29_1_179_0
ER  -

%0 Journal Article
%A Denis Koleda
%T On the density function of the distribution of real algebraic numbers
%J Journal de théorie des nombres de Bordeaux
%D 2017
%P 179-200
%V 29
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.975/
%R 10.5802/jtnb.975
%G en
%F JTNB_2017__29_1_179_0

Denis Koleda. On the density function of the distribution of real algebraic numbers. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 179-200. doi : 10.5802/jtnb.975. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.975/

Bibliographie
Cité par

[1] P. Bachmann Zahlentheorie. II. Theil. Die analytische Zahlentheorie, BG Teubner, Leipzig, 1894, xviii+494 pages

[2] A. Baker; W. M. Schmidt Diophantine approximation and Hausdorff dimension, Proc. Lond. Math. Soc., Volume 21 (1970), pp. 1-11 | DOI

[3] V. Beresnevich On approximation of real numbers by real algebraic numbers, Acta Arith., Volume 90 (1999) no. 2, pp. 97-112

[4] V. I. Bernik Use of Hausdorff dimension in the theory of Diophantine approximations, Acta Arith., Volume 42 (1983) no. 3, pp. 219-253 (in Russian)

[5] H. Brown; K. Mahler A generalization of Farey sequences: Some exploration via the computer, J. Number Theory, Volume 3 (1971) no. 3, pp. 364-370 | DOI

[6] Y. Bugeaud Approximation by algebraic numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press, 2004, xvi+274 pages

[7] C. Cobeli; A. Zaharescu The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform., Volume 5 (2003), pp. 1-38

[8] H. Davenport On a principle of Lipschitz, J. Lond. Math. Soc., Volume 26 (1951), pp. 179-183 Corrigendum: “On a principle of Lipschitz”, J. Lond. Math. Soc. 39 (1964), p. 580 | DOI

[9] F. Dress Discrépance des suites de Farey, J. Théor. Nombres Bordeaux, Volume 11 (1999) no. 2, pp. 345-367 | DOI

[10] A. Dubickas On the number of reducible polynomials of bounded naive height, Manuscripta Mathematica, Volume 144 (2014) no. 3–4, pp. 439-456 | DOI

[11] A. Edelman; E. Kostlan How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc., Volume 32 (1995) no. 1, pp. 1-37 Erratum: Bull. Amer. Math. Soc. 33 (1996), no. 3, p. 325 | DOI

[12] P. Erdős; M. Kac; E. R. van Kampen; A. Wintner Ramanujan sums and almost periodic functions, Studia Math., Volume 9 (1940), pp. 43-53

[13] P. Erdős; P. Turán On the distribution of roots of polynomials, Annals of Mathematics, Volume 51 (1950), pp. 105-119 | DOI

[14] J. Franel Les suites de Farey et le problème des nombres premiers, Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl. (1924), pp. 198-201

[15] I. Ibragimov; D. Zaporozhets On distribution of zeros of random polynomials in complex plane, Prokhorov and Contemporary Probability Theory (Springer Proc. Math. Stat.), Volume 33, Springer, 2013, pp. 303-323

[16] I. A. Ibragimov; N. B. Maslova On the expected number of real zeros of random polynomials I. Coefficients with zero means, Teor. Verojatnost. i Primenen., Volume 16 (1971) no. 2, pp. 229-248 in Russian, English translation in Theor. Probability Appl. 16 (1971), no. 2, p. 228-248

[17] M. Kac On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc., Volume 49 (1943) no. 4, pp. 314-320 Corrigendum: Bull. Amer. Math. Soc. 49 (1943), no. 12, p. 938 | DOI

[18] M. Kac On the average number of real roots of a random algebraic equation II, Proc. Lond. Math. Soc., Volume 50 (1948), pp. 390-408 | DOI

[19] D. Kaliada Distribution of real algebraic numbers of a given degree, Dokl. Nats. Akad. Nauk Belarusi, Volume 56 (2012) no. 3, pp. 28-33 (in Belarusian)

[20] D. Kaliada Distribution of real algebraic numbers of the second degree, Vestsi Nats. Akad. Navuk Belarusi Ser. fiz.-mat. navuk (2013) no. 3, pp. 54-63 (in Russian)

[21] D. Koleda On the asymptotic distribution of algebraic numbers with growing naive height, Chebyshevskii Sbornik, Volume 16 (2015) no. 1, pp. 191-204 (in Russian)

[22] L. Kuipers; H. Niederreiter Uniform distribution of sequences, Pure and Applied Mathematics, John Wiley & Sons, 1974, xiv+390 pages

[23] D. Masser; J. D. Vaaler Counting algebraic numbers with large height II, Trans. Amer. Math. Soc., Volume 359 (2007) no. 1, pp. 427-445 | DOI

[24] D. Masser; J. D. Vaaler Counting algebraic numbers with large height I, Diophantine approximation (Dev. Math.), Volume 16, Springer, 2008, pp. 237-243

[25] M. Mikolás Farey series and their connection with the prime number problem. I, Acta Sci. Math. (Szeged), Volume 13 (1949), pp. 93-117

[26] H. Niederreiter The distribution of Farey points, Math. Ann., Volume 201 (1973), pp. 341-345 | DOI

[27] V. V. Prasolov Polynomials, Algorithms and Computation in Mathematics, 11, Springer, Berlin, 2004, xiv+301 pages (translated from the 2001 Russian second edition by Dimitry Leites)

[28] B. L. van der Waerden Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt, Monatsh. Math. Phys., Volume 43 (1936) no. 1, pp. 133-147 | DOI

[29] B. L. van der Waerden Algebra I, Heidelberger Taschenbücher, 12, Springer-Verlag, Berlin-Heidelberg, 1971, ix+272 pages

[30] A. Walfisz Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, 15, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963, 231 pages

[31] D. N. Zaporozhets Random polynomials and geometric probability, Doklady Mathematics, Volume 71 (2005) no. 1, pp. 53-57

Cité par Sources :