Intervals of small measure containing an algebraic number of given height

N. I. Kalosha ; I. A. Korlyukova ; E. V. Guseva

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 213-220.

Résumé

Rational numbers are uniformly distributed, even though distances between rational neighbors in a Farey sequence can be quite different. This property doesn't hold for algebraic numbers. In 2013 D. Koleda [6, 7] found the distribution function for real algebraic numbers of an arbitrary degree under their natural ordering. It can be proved that the quantity of real algebraic numbers $ \alpha $ of degree $n$ and height $H( \alpha ) \le Q$ asymptotically equals $c_{1}(n)Q^{n+1}$. Recently it was proved that there exist intervals of length $Q^{- \gamma }, \gamma >1$, free of algebraic numbers $ \alpha , H( \alpha ) \le Q$, however for $0 \le \gamma 1$ there exist at least $c_{2}(n)Q^{n+1- \gamma }$ algebraic numbers in such intervals. In this paper we show that special intervals of length $Q^{- \gamma }$ may contain algebraic numbers even for large values of $ \gamma $, however their quantity doesn't exceed $c_{3}Q^{n+1- \gamma }$. An earlier result by A. Gusakova [16] was proved only for the case $\gamma = \frac{3}{2}$.

Détail

Keywords: algebraic number, Diophantine approximation, uniform distribution, Dirichlet's theorem.

Bibliographie

[1] Bernik V. I., Tischenko K. I., “Tselochislennye mnogochleny s perepadami vysot koeffitsientov i gipoteza Virzinga”, Dokl. AN Belarusi, 37:5 (1993), 9–11 | MR | Zbl

[2] Bernik V. I., Gettse F., “Raspredelenie deistvitelnykh algebraicheskikh chisel proizvolnoi stepeni v korotkikh intervalakh”, Izv. RAN Ser. matem., 79:1 (2014), 21–42

[3] Bernik V. I., Gettse F., “Raspredelenie deistvitelnykh algebraicheskikh chisel proizvolnoi stepeni v korotkikh intervalakh”, ispravlennyi variant, Izv. RAN. Ser. matem., 79:1 (2015), 21–42 | MR | Zbl

[4] Gelfond A. O., UMN, 4:5 (1949), 14–48 | MR | Zbl

[5] Kassels Dzh. V., Vvedenie v teoriyu diofantovykh priblizhenii, Izd-vo Inostr. Liter., M., 1961, 213 pp.

[6] Koleda D. V., “O raspredelenii deistvitelnykh algebraicheskikh chisel vtoroi stepeni”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 2013, no. 3, 54–63

[7] D. V. Koleda, “On the density function of the distribution of real algebraic numbers”, J. Théor. Nombres Bordeaux, 29:1 (2017), 179–200 | DOI | MR | Zbl

[8] Sprindzhuk V. G., Problema Malera v metricheskoi teorii chisel, Nauka i tekhnika, Minsk, 1967, 181 pp. | MR

[9] V. Beresnevich, “Rational Points near Manifolds and Metric Diophantine Approximation”, Annals of Mathematics, Second Series, 175:1 (2012), 187–235 | DOI | MR | Zbl

[10] V. V. Beresnevich, V. I. Bernik, F. Goetze, “Integral polynomials with small discriminants and resultants”, Adv. Math., 298 (2016), 393–412 | DOI | MR | Zbl

[11] Bernik V. I., “Primenenie razmernosti Khausdorfa v teorii diofantovykh priblizhenii”, Acta Arithmetica, 42 (1983), 219–253 | DOI | Zbl

[12] M. M. Dodson, V. I. Bernik, Metric Diophantine approximation on manifolds, Cambridge Tracts in Mathematics, 137, 1999, 172 pp. | MR | Zbl

[13] K. Tishchenko, “On approximation to real numbers by algebraic numbers”, Acta Arithmetica, 94:1 (2000), 1–24 | DOI | MR | Zbl

[14] H. Davenport, W. M. Schmidt, “Dirichlet's theorem on diophantine approximation”, Symposia Mathematica, 4, N.Y.–London, 1970, 113–132 | MR | Zbl

[15] E. Wirsing, “Approximation mit algebraischen Zahlen beschränkten Grades”, Journal für die reine und angewandte Mathematik, 206 (1961), 67–77 | MR | Zbl

[16] Gettse F., Gusakova A. G., “Algebraicheskie chisla v korotkikh intervalakh”, Doklady Natsionalnoi akademii nauk Belarusi, 59:4 (2015), 11–16 | MR

[17] D. Badziahin, J. Schleischitz, An improved bound in Wirsing's problem, arXiv: abs/1912.09013