Contribution of Jonas Kubilius to the metric theory of diophantine approximation of dependent variables

V. V. Beresnevich ; V. I. Bernik ; F. Götze ; E. V. Zasimovich ; N. I. Kalosha

Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2021), pp. 34-50

Résumé

The article is devoted to the latest results in metric theory of Diophantine approximation. One of the first major result in area of number theory was a theorem by academician Jonas Kubilius. This paper is dedicated to centenary of his birth. Over the last 70 years, the area of Diophantine approximation yielded a number of significant results by great mathematicians, including Fields prize winners Alan Baker and Grigori Margulis. In 1964 academician of the Academy of Sciences of BSSR Vladimir Sprindžuk, who was a pupil of academician J. Kubilius, solved the well-known Mahler’s conjecture on the measure of the set of S-numbers under Mahler’s classification, thus becoming the founder of the Belarusian academic school of number theory in 1962.

Détail

Keywords: J. Kubilius; Diophantine approximation; Mahler’s conjecture; metric number theory; transcendence and algebraic numbers.

Bibliographie

[1] I. P. Kubilyus, “O primenenii metoda akademika Vinogradova k resheniyu odnoi zadachi metricheskoi teorii chisel”, Doklady Akademii nauk SSSR, 67 (1949), 783–786 | MR | Zbl

[2] I. P. Kubilyus, “Ob odnoi metricheskoi probleme teorii diofantovykh priblizhenii”, Doklady Akademii nauk Litovskoi SSR, 2 (1959), 3–7

[3] V. G. Sprindzhuk, “Dokazatelstvo gipotezy Malera o mere mnozhestva S-chisel”, Izvestiya Akademii nauk SSSR, 29(2) (1965), 379–436 | Zbl

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

[5] V. G. Sprindzhuk, “Dostizheniya i problemy teorii diofantovykh priblizhenii”, Uspekhi matematicheskikh nauk, 35(4) (1980), 3–68 | Zbl

[6] M. E. Borel, “Les probabilites denombrables et leurs applications arithmetiques”, Rendiconti del Circolo Matematico di Palermo (1884–1940), 27 (1909), 247–271 | DOI | Zbl

[7] A. Khintchine, “Einige Satze uber Kettenbruche, mit Anwendungen auf die Theorie der diophantischen Approximationen”, Mathematische Annalen, 92 (1924), 115–125 | DOI | MR | Zbl

[8] A. V. Groshev, “Teorema o sisteme lineinykh form”, Doklady Akademii nauk SSSR, 19 (1938), 151–152

[9] G. Harman, Metric Number Theory, London Mathematical Society monographs, 18, Clarendon Press, Oxford, 1998 | MR | Zbl

[10] JWS. Cassels, An introduction to Diophantine approximation, Cambridge tracts in mathematics and mathematical physics, Cambridge University Press, Cambridge, 1957 | MR

[11] V. G. Sprindzuk, “Mahler’s problem in metric number theory”, Translations of mathematical monographs, 25, American Mathematical Society, Providence, 1969, 192 | MR

[12] A. Khintchine, “Zwei Bemerkungen zu einer Arbeit des Herrn Perron”, Mathematische Zeitschrift, 22 (1925), 274–284 | DOI | MR | Zbl

[13] K. Mahler, “Uber das Mab der Menge aller S-Zahlen”, Mathematische Annalen, 106 (1932), 131–139 | DOI | MR

[14] W. M. Schmidt, “Bounds for certain sums; a remark on a conjecture of Mahler”, Transactions of the American Mathematical Society, 101(2) (1961), 200–210 | DOI | MR | Zbl

[15] Y. Bugeaud, “Approximation by algebraic numbers”, Cambridge tracts in mathematics, 160, Cambridge University Press, Cambridge, 2004, 290 | DOI | MR

[16] J. F. Koksma, “Uber die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen”, Monatshefte fur Mathematik, 48(1) (1939), 176–189 | DOI | MR

[17] W. J. LeVeque, “Note on S-numbers”, Proceedings of the American Mathematical Society, 4 (1953), 189–190 | DOI | MR | Zbl

[18] F. Kasch, B. Volkmann, “Zur Mahlerschen Vermutung ber S-Zahlen”, Mathematische Annalen, 136(5) (1958), 442–453 | DOI | MR | Zbl

[19] W. M. Schmidt, “Metrische Satze uber simultane Approximation abhangiger Groben”, Monatshefte fur Mathematik, 68(2) (1964), 154–166 | DOI | MR | Zbl

[20] B. Volkmann, “The real cubic case of Mahler’s conjecture”, Mathematika, 8(1) (1961), 55–57 | DOI | MR | Zbl

[21] H. Davenport, “A note on binary cubic forms”, Mathematika, 8(1) (1961), 58–62 | DOI | MR | Zbl

[22] V. Bernik, F. Gotze, A. Gusakova, “On points with algebraically conjugate coordinates close to smooth curves”, Zapiski nauchnykh seminarov POMI, 448 (2016), 14–47 | MR

[23] JWS. Cassels, “Some metrical theorems in Diophantine approximation: v on a conjecture of Mahler”, Mathematical Proceedings of the Cambridge Philosophical Society, 47(1) (1951), 18–21 | DOI | MR

[24] A. Baker, W. M. Schmidt, “Diophantine approximation and Hausdorff dimension”, Proceedings of the London Mathematical Society, s3-21(1) (1970), 1–11 | DOI | MR | Zbl

[25] V. Bernik, “O tochnom poryadke priblizheniya nulya znacheniyami tselochislennykh mnogochlenov”, Acta Arithmetica, 53(1) (1989–1990), 17–28 | DOI | Zbl

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

[27] V. Beresnevich, “On a theorem of V Bernik in the metric theory of Diophantine approximation”, Acta Arithmetica, 117(1) (2005), 71–80 | DOI | MR | Zbl

[28] V. I. Bernik, D. V. Vasilev, “Teorema tipa Khinchina dlya tselochislennykh mnogochlenov kompleksnoi peremennoi”, Trudy Instituta matematiki NAN Belarusi, 3 (1999), 10–20 | Zbl

[29] V. V. Beresnevich, V. I. Bernik, E. I. Kovalevskaya, “On approximation of p-adic numbers by p-adic algebraic numbers”, Journal of Number Theory, 111(1) (2005), 33–56 | DOI | MR | Zbl

[30] A. Mohammadi, A. S. Golsefidy, “S-arithmetic Khintchine-type theorem”, Geometric and Functional Analysis, 19(4) (2009), 1147–1170 | DOI | MR | Zbl

[31] F. Adiceam, V. Beresnevich, J. Levesley, S. Velani, E. Zorin, “Diophantine approximation and applications in interference alignment”, Advances in Mathematics, 302 (2016), 231–279 | DOI | MR | Zbl

[32] V. I. Bernik, N. V. Shamukova, “Priblizhenie deistvitelnykh chisel tselymi algebraicheskimi chislami i teorema Khinchina”, Doklady Natsionalnoi akademii nauk Belarusi, 50(3) (2006), 30–32 | Zbl

[33] V. I. Bernik, M. M. Dodson, “Metric Diophantine approximation on manifolds”, Cambridge tracts in mathematics, 137, Cambridge University Press, Cambridge, 1999, 172 | MR | Zbl

[34] V. Beresnevich, V. Bernik, “On a metrical theorem of W Schmidt”, Acta Arithmetica, 75(3) (1996), 219–233 | DOI | MR | Zbl

[35] D. Y. Kleinbock, G. A. Margulis, “Flows on homogeneous spaces and Diophantine approximation on manifolds”, Annals of Mathematics, 148(1) (1998), 339–360 | DOI | MR | Zbl

[36] V. Beresnevich, “A Groshev type theorem for convergence on manifolds”, Acta Mathematica Hungarica, 94(1–2) (2002), 99–130 | DOI | MR | Zbl

[37] V. Bernik, D. Kleinbock, G. Margulis, “Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions”, International Mathematics Research Notices, 2001(9) (2001), 453–486 | DOI | MR | Zbl

[38] V. V. Beresnevich, V. I. Bernik, D. Y. Kleinbock, G. A. Margulis, “Metric Diophantine approximation: the Khintchine – Groshev theorem for non-degenerate manifolds”, Moscow Mathematical Journal, 2(2) (2002), 203–225 | DOI | MR | Zbl

[39] A. Baker, “On a theorem of Sprindzuk”, Proceedings of the Royal Society of London, Mathematical and Physical Sciences, 292(1428), 1966, 92–104 | DOI | Zbl

[40] J. F. Koksma, “Diophantische approximationen”, Ergebnisse der Mathematik und ihrer Grenzgebiete, 4, Berlin: Springer, 1974, 172 | DOI | MR

[41] D. Allen, V. Beresnevich, “A mass transference principle for systems of linear forms and its applications”, Compositio Mathematica, 154(5) (2018), 1014–1047 | DOI | MR | Zbl

[42] V. I. Bernik, D. V. Vasilev, E. V. Zasimovich, “Diofantovy priblizheniya s postoyannoi pravoi chastyu neravenstv na korotkikh intervalakh”, Doklady Natsionalnoi akademii nauk Belarusi, 65(4) (2021), 397–403 | DOI

[43] V. Jarnik, “Diophantische approximationen und Hausdorffsches mass”, Matematicheskii sbornik', 36(3–4) (1929), 371–382 | Zbl

[44] A. S. Besicovitch, “Sets of fractional dimensions (IV): on rational approximation to real numbers”, Journal of the London Mathematical Society, s1-9(2) (1934), 126–131 | DOI | MR

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

[46] Yu. V. Melnichuk, “Diofantovy priblizheniya na okruzhnosti i razmernost Khausdorfa”, Matematicheskie zametki, 26(3) (1979), 347–354

[47] V. I. Bernik, I. L. Morotskaya, “Diofantovy priblizheniya v Q (s ind. p) i razmernost Khausdorfa”, Vestsi Natsyyanalnai akademii navuk Belarusi, 3 (1986), 3–9 | MR | Zbl

[48] V. I. Bernik, N. I. Kalosha, “Priblizhenie nulya znacheniyami tselochislennykh polinomov v prostranstve R * C * Q (s ind. p)”, Vestsi Natsyyanalnai akademii navuk Belarusi, 1 (2004), 121–123

[49] N. V. Budarina, “Metricheskaya teoriya sovmestnykh diofantovykh priblizhenii v R(v step. k)*C*Q(v step l, s ind. p)”, Chebyshevskii sbornik. Posvyaschaetsya 65-i godovschine so dnya rozhdeniya professora Sergeya Mikhailovicha Voronina, 12(1) (2011), 17–50 | Zbl

[50] D. Badziahin, J. Schleischitz, “An improved bound in Wirsing’s problem”, Transactions of the American Mathematical Society, 374 (2021), 1847–1861 | DOI | MR | Zbl

[51] V. I. Bernik, K. I. Tischenko, “Tselochislennye mnogochleny s perepadami vysot koeffitsientov i gipoteza Virzinga”, Doklady Natsionalnoi akademii nauk Belarusi, 37(5) (1993), 9–11 | Zbl

[52] V. Beresnevich, D. Dickinson, S. Velani, “Diophantine approximation on planar curves and the distribution of rational points”, Annals of Mathematics, 166(2) (2007), 367–426 | DOI | MR | Zbl

[53] H. Dickinson, M. M. Dodson, “Extremal manifolds and Hausdorff dimension”, Duke Mathematical Journal, 101(2) (2000), 271–281 | DOI | MR | Zbl

[54] B. P. Rynne, “Simultaneous Diophantine approximation on manifolds and Hausdorff dimension”, Journal of Number Theory, 98(1) (2003), 1–9 | DOI | MR | Zbl

[55] A. S. Kudin, A. V. Lunevich, “Analog teoremy Khinchina v sluchae raskhodimosti v polyakh deistvitelnykh, kompleksnykh i p-adicheskikh chisel”, Trudy Instituta matematiki, 23(1) (2015), 76–83 | Zbl

[56] V. Beresnevich, E. Zorin, “Explicit bounds for rational points near planar curves and metric Diophantine approximation”, Advances in Mathematics, 225(6) (2010), 3064–3087 | DOI | MR | Zbl

[57] V. Beresnevich, R. C. Vaughan, S. Velani, E. Zorin, “Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory”, International Mathematics Research Notices, 2017(10) (2017), 2885–2908 | DOI | MR | Zbl

[58] J-J. Huang, “The density of rational points near hypersurfaces”, Duke Mathematical Journal, 169(11) (2020), 2045–2077 | DOI | MR | Zbl

[59] D. Simmons, “Some manifolds of Khinchin type for convergence”, Journal de Theorie des Nombres de Bordeaux, 30(1) (2018), 175–193 | DOI | MR | Zbl

[60] V. Beresnevich, S. Velani, “An inhomogeneous transference principle and Diophantine approximation”, Proceedings of the London Mathematical Society, 101(3) (2010), 821–851 | DOI | MR | Zbl

[61] K. Yu. Yavid, “Otsenka razmernosti Khausdorfa mnozhestv singulyarnykh vektorov”, Doklady Akademii nauk BSSR, 31(9) (1987), 777–780 | MR | Zbl

[62] V. Beresnevich, J. Levesley, B. Ward, “A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds”, International Journal of Number Theory, 17(8) (2021), 1795–1814 | DOI | MR | Zbl

[63] V. I. Bernik, “Applications of measure theory and Hausdorff dimension to the theory of Diophantine approximation”, New advances in transcendence theory, 1988, 25–36 | DOI | MR | Zbl

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

[65] V. Beresnevich, L. Lee, R. C. Vaughan, S. Velani, “Diophantine approximation on manifolds and lower bounds for Hausdorff dimension”, Mathematika, 63(3) (2017), 762–779 | DOI | MR | Zbl

[66] V. I. Bernik, N. A. Pereverseva, “The method of trigonometric sums and lower estimates of Hausdorff dimension”, Analytic and Probabilistic Methods in Number Theory, 2 (1992), 75–81 | DOI | MR | Zbl

[67] Y. Bugeaud, “Approximation by algebraic integers and Hausdorff dimension”, Journal of the London Mathematical Society, 65(3) (2002), 547–559 | DOI | MR | Zbl

[68] V. V. Beresnevich, S. L. Velani, “A note on simultaneous Diophantine approximation on planar curves”, Mathematische Annalen, 337(4) (2007), 769–796 | DOI | MR | Zbl

[69] V. V. Beresnevich, “Primenenie ponyatiya regulyarnykh sistem tochek v metricheskoi teorii chisel”, Vestsi Natsyyanalnai akademii navuk Belarusi, 1 (2000), 35–39

[70] V. V. Beresnevich, “O postroenii regulyarnykh sistem tochek s veschestvennymi, kompleksnymi i p-adicheskimi algebraicheskimi koordinatami”, Vestsi Natsyyanalnai akademii navuk Belarusi, 1 (2003), 22–27

[71] E. I. Kovalevskaya, V. Bernik, “Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations”, Acta Mathematica Universitatis Ostraviensis, 14(1) (2006), 37–42 | MR | Zbl

[72] V. І. Bernik, І. M. Marozava, “Gipoteza Beikera i regulyarnyya sistemy algebraichnykh lichbau z abmezhavannem na znachenne vytvornai”, Vestsi Natsyyanalnai akademii navuk Belarusi, 3 (1996), 109–113 | Zbl

[73] D. Kleinbock, G. Tomanov, “Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation”, Commentarii Mathematici Helvetici, 82(3) (2007), 519–581 | DOI | MR | Zbl

[74] V. I. Bernik, N. V. Sakovich, “Regulyarnye sistemy kompleksnykh algebraicheskikh chisel”, Doklady Akademii nauk Belarusi, 38(5) (1994), 10–13 | MR | Zbl

[75] V. V. Beresnevich, E. I. Kovalevskaya, “O diofantovykh priblizheniyakh zavisimykh velichin v p-adicheskom sluchae”, Matematicheskie zametki, 25(1) (2003), 22–37 | DOI

[76] V. I. Bernik, “Metricheskaya teorema o sovmestnom priblizhenii nulya znacheniyami tselochislennykh mnogochlenov”, Izvestiya Akademii nauk SSSR, 44(1) (1980), 24–45 | Zbl

[77] F. Zeludevic, “Simultane diophantische Approximationen abhangiger Groben in mehreren Metriken”, Acta Arithmetica, 46(3) (1986), 285–296 | DOI | MR

[78] V. Bernik, N. Budarina, D. Dickinson, “A divergent Khintchine theorem in the real, complex, and p-adic fields”, Lithuanian Mathematical Journal, 48(2) (2008), 158–173 | DOI | MR | Zbl

[79] N. Budarina, D. Dickinson, V. Bernik, “Simultaneous Diophantine approximation in the real, complex and p-adic fields”, Mathematical Proceedings of the Cambridge Philosophical Society, 149(2) (2010), 193–216 | DOI | MR | Zbl

[80] I. R. Dombrovskii, “Sovmestnye priblizheniya deistvitelnykh chisel algebraicheskimi chislami ogranichennoi stepeni”, Doklady Akademii nauk BSSR, 33(3) (1989), 205–208

[81] V. Beresnevich, “Rational points near manifolds and metric Diophantine approximation”, Annals of Mathematics, 175(1) (2012), 187–235 | DOI | MR | Zbl

[82] V. V. Beresnevich, S. L. Velani, “Simultaneous inhomogeneous Diophantine approximation on manifolds”, Journal of Mathematical Sciences, 180(5) (2012), 531–541 | DOI | MR | Zbl

[83] N. Budarina, D. Dickinson, J. Levesley, “Simultaneous Diophantine approximation on polynomial curves”, Mathematika, 56(1) (2010), 77–85 | DOI | MR | Zbl

[84] N. Budarina, “On a problem of Bernik, Kleinbock and Margulis”, Glasgow Mathematical Journal, 53(3) (2011), 669–681 | DOI | MR | Zbl

[85] N. Budarina, D. Dickinson, “Simultaneous Diophantine approximation in two metrics and the distance between conjugate algebraic numbers in R*Q (v step. p)”, Indagationes Mathematicae, 23(1–2) (2012), 32–41 | DOI | MR | Zbl

[86] V. I. Bernik, V. N. Borbat, “Sovmestnaya approksimatsiya nulya znacheniyami tselochislennykh polinomov”, Trudy Matematicheskogo instituta imeni VA Steklova, 218 (1997), 58–73 | Zbl

[87] V. I. Bernik, N. Budarina, H. O’Donnell, “On regular systems of real algebraic numbers of third degree in short intervals”, Sovremennye problemy matematiki, 17 (2013), 61–75 | DOI | MR | Zbl

[88] Y. Bugeaud, M. Mignotte, “Polynomes a coefficients entiers prenant des valeurs positives aux points reels”, Bulletin mathematique de la Societe des Sciences Mathematiques de Roumanie, 53(3) (2010), 219–224 | MR

[89] V. Bernik, F. Gotze, “A new connection between metric theory of Diophantine approximations and distribution of algebraic numbers”, Contemporary Mathematics, 631 (2015), 33–45 | DOI | MR | Zbl

[90] N. V. Budarina, “Sovmestnye diofantovy priblizheniya s nemonotonnymi pravymi chastyami”, Doklady Akademii nauk, 437(4) (2011), 441–443 | Zbl

[91] V. Bernik, Guire. Mc, “How small can polynomials be in an interval of given length”, Glasgow Mathematical Journal, 62(2) (2020), 261–280 | DOI | MR | Zbl

[92] Y. Bugeaud, M. Mignotte, “Polynomial root separation”, International Journal of Number Theory, 6(3) (2010), 587–602 | DOI | MR | Zbl

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

[94] V. Bernik, N. Budarina, H. O’Donnell, “Discriminants of polynomials in the Archimedean and non-Archimedean metrics”, Acta Mathematica Hungarica, 154(2) (2018), 265–278 | DOI | MR | Zbl

[95] M. N. Huxley, “The rational points close to a curve. Annali della Scuola Normale Superiore di Pisa”, Classe di Scienze, 21(3) (1994), 357–375 | MR | Zbl

[96] D. V. Koleda, “Ob asimptotike raspredeleniya algebraicheskikh chisel pri vozrastanii ikh vysot”, Chebyshevskii sbornik, 16(1) (2015), 191–204 | DOI | MR | Zbl

[97] F. Gotze, D. Koleda, D. Zaporozhets, “Joint distribution of conjugate algebraic numbers: a random polynomial approach”, Advances in Mathematics, 359 (2020), 106849 | DOI | MR | Zbl

[98] V. V. Lebed, V. I. Bernik, “Algebraicheskie tochki na ploskosti”, Fundamentalnaya i prikladnaya matematika, 11(6) (2005), 73–80

[99] D. V. Koleda, “O raspredelenii veschestvennykh algebraicheskikh chisel vtoroi stepeni”, Vestsi Natsyyanalnai akademii navuk Belarusi, 3 (2013), 54–63

[100] D. Koleda, “On the density function of the distribution of real algebraic numbers”, Journal de Theorie des Nombres de Bordeaux, 29(1) (2017), 179–200 | DOI | MR | Zbl

[101] V. I. Bernik, F. Gettse, N. I. Kalosha, “O kolichestve algebraicheskikh chisel v korotkikh intervalakh, soderzhaschikh ratsionalnye tochki”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 1 (2019), 4–11 | DOI | Zbl

[102] N. V. Budarina, D. Dikkinson, V. I. Bernik, “Otsenki snizu dlya kolichestva vektorov s algebraicheskimi koordinatami vblizi gladkikh poverkhnostei”, Doklady Natsionalnoi akademii nauk Belarusi, 64(1) (2020), 7–12 | DOI

[103] V. I. Bernik, F. Gotze, “Distribution of real algebraic numbers of arbitrary degree in short intervals”, Izvestiya: Mathematics, 79(1) (2015), 18–39 | DOI | MR | Zbl

[104] N. Budarina, D. Dickinson, “Diophantine approximation on non-degenerate curves with non-monotonic error function”, Bulletin of the London Mathematical Society, 41(1) (2009), 137–146 | DOI | MR | Zbl

[105] N. Budarina, “Diophantine approximation on the curves with non-monotonic error function in the p-adic case”, Chebyshevskii sbornik, 11(1) (2010), 74–80 | MR | Zbl

[106] D. Badziahin, V. Beresnevich, S. Velani, “Inhomogeneous theory of dual Diophantine approximation on manifolds”, Advances in Mathematics, 232(1) (2013), 1–35 | DOI | MR | Zbl

[107] D. Badziahin, “Inhomogeneous Diophantine approximation on curves and Hausdorff dimension”, Advances in Mathematics, 223(1) (2010), 329–351 | DOI | MR | Zbl

[108] V. Bernik, H. Dickinson, J. Yuan, “Inhomogeneous Diophantine approximation on polynomials in Q (v step. p)”, Acta Arithmetica, 90(1) (1999), 37–48 | DOI | MR | Zbl

[109] V. Beresnevich, A. Ganguly, A. Ghosh, S. Velani, “Inhomogeneous dual Diophantine approximation on affine subspaces”, International Mathematics Research Notices, 12 (2020), 3582–3613 | DOI | MR | Zbl

[110] V. V. Beresnevich, R. C. Vaughan, S. L. Velani, “Inhomogeneous Diophantine approximation on planar curves”, Mathematische Annalen, 349(4) (2011), 929–942 | DOI | MR | Zbl

[111] V. I. Arnold, “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, Uspekhi matematicheskikh nauk, 18(6) (1963), 91–192 | Zbl

[112] B. I. Ptashnik, “Nekorrektnye granichnye zadachi dlya differentsialnykh uravnenii s chastnymi proizvodnymi”, Kiev: Naukova dumka, 1984, 264 | MR

[113] V. R. Cadambe, S. A. Jafar, “Interference alignment and degrees of freedom of the K-user interference channel”, IEEE Transactions on Information Theory, 54(8) (2008), 3425–3441 | DOI | MR | Zbl

[114] S. A. Jafar, “Interference alignment: a new look at signal dimensions in a communication network”, Foundations and Trends in Communications and Information Theory, 7(1) (2011), 1–134 | DOI

[115] S. A. Jafar, S. h. Shamai, “Degrees of freedom region of the MIMO X channel”, IEEE Transactions on Information Theory, 54(1) (2008), 151–170 | DOI | MR | Zbl

[116] V. Beresnevich, S. Velani, “Number theory meets wireless communications: an introduction for dummies like u. Number theory meets wireless communications”, Springer International Publishing, 2020, 1–67 | DOI | MR | Zbl

[117] N. Budarina, H. O’Donnell, “On a problem of Nesterenko: when is the closest root of a polynomial a real number”, International Journal of Number Theory, 8(3) (2012), 801–811 | DOI | MR | Zbl

[118] A. S. Motahari, S. Oveis-Gharan, M. A. Maddah-Ali, A. K. Khandani, “Real interference alignment: exploiting the potential of single antenna systems”, IEEE Transactions on Information Theory, 60(8) (2014), 4799–4810 | DOI | MR | Zbl

[119] R. Slesoraitene, “Teorema Malera – Sprindzhuka dlya polinomov tretei stepeni ot dvukh peremennykh”, Litovskii matematicheskii sbornik, 10 (1970), 791–814 | MR

[120] E. Manstavičius, “Jonas Kubilius 1921–2011”, Lithuanian Mathematical Journal, 61 (2021), 285–288 | DOI | MR | Zbl