Approximation of real numbers by

V. I. Bernik ; E. V. Guseva ; N. V. Sakovich

Trudy Instituta matematiki, Tome 28 (2020) no. 1, pp. 3-10.

Résumé

The problem of approximating real numbers by algebraic numbers of a given degree and height is a natural development of the classical Dirichlet's theorem from the mid-19th century, which described approximation of real numbers by rational fractions. Approximation by algebraic numbers was first studied in 1961 by a German mathematician E. Wirsing. This article describes contributions of Belarusian mathematicians V. Sprindzuk, V. Bernik, K. Tishchenko, V. Beresnevich, D. Koleda, A. Gusakova, and D. Bodyagin to the research related to Wirsing's conjecture, as well as studies of the distribution of algebraic numbers, their discriminants and resultants. In addition, a conjecture of V. Beresnevich, V. Bernik and F. Goetze has been proved.

Bibliographie

[1] Bernik V. I., “O tochnom poryadke priblizheniya nulya znacheniyami tselochislennykh mnogochlenov”, Acta Arith., 53:1 (1989), 17–28 | DOI | MR | Zbl

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

[3] Kalyada D. U., “Ab razmerkavanni rechaisnykh algebraichnykh likaŭ dadzenai stupeni”, Dokl. NAN Belarusi, 56:3 (2012), 28–33 | MR | Zbl

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

[5] Shmidt V., Diofantovy priblizheniya, Per. s angl., Mir, M., 1983, 232 pp. | MR

[6] Badziahin D., Schleischitz J., An improved bound in Wirsing's problem, 2019, arXiv: 1912.09013v1 | MR

[7] Beresnevich V., “On approximation of real numbers by real algebraic numbers”, Acta Arith., 90:2 (1999), 97–112 | DOI | MR | Zbl

[8] Beresnevich V., Bernik V., Goetze F., “Integral polynomials with small discriminants and resultants”, Advances in Mathematics, 2016, 393–412 | DOI | MR | Zbl

[9] Bernik V. I., Dodson M. M., Metric Diophantine Approximation on Manifolds, Cambridge, U.K, 1991 | MR

[10] Bernik V. I., Tsishchanka K., “Integral polynomials with an overfall of the coefficient values and Wirsing's theorem”, Dokl. Akad. Nauk Belarusi, 37:5 (1993), 9–11 (Russian) ; (1994), 121 | MR | Zbl

[11] Davenport H., Schmidt W. M., “Approximation to real numbers by algebraic integers”, Acta Arith., 15 (1969), 393–416 | DOI | MR | Zbl

[12] Khintchine A., “Über die angenäherte Auflösung linearer Gleichungen in ganzen Zahlen”, Matem. sb., 32:1 (1924), 203–219

[13] Tsishchanka K. I., “On approximation of real numbers by algebraic numbers of bounded degree”, J. Number Theory, 123 (2007), 290–314 | DOI | MR | Zbl

[14] Wirsing E., “Approximation mit algebraischen Zahlen beschrdnkten Grades”, J. Reine Angew. Math., 206 (1961), 67–77 | DOI | MR | Zbl

[15] Beresnevich V., Bernik V., Götze F., “The distribution of close conjugate algebraic numbers”, Compositio Mathematica, 146:5 (2010), 1165–1179 | DOI | MR | Zbl

[16] Bernik V. I., Vasilev D. V., Kudin A. S., “O chisle tselochislennykh mnogochlenov zadannoi stepeni i ogranichennoi vysoty s maloi proizvodnoi v korne mnogochlena”, Trudy Instituta matematiki, 22:2 (2014), 3–8 | Zbl

[17] Baker A., Schmidt W. M., “Diophantine approximation and Hausdorff dimension”, Proceedings of the London Mathematical Society, 3:1 (1970), 1–11 | DOI | MR | Zbl

[18] Baker R. C., “On approximation with algebraic numbers of bounded degree”, Mathematika, 23:1 (1976), 18–31 | DOI | MR | Zbl

[19] Bernik V. I., “Primenenie razmernosti Khausdorfa v teorii diofantovykh priblizhenii”, Acta Arithmetica, 42:3 (1983), 219–253 | DOI | MR | Zbl