Quantitative versions of the Subspace Theorem and applications
Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, p. 35-57

During the last decade, several quite unexpected applications of the Schmidt Subspace Theorem were found. We survey some of these, with a special emphasize on the consequences of quantitative statements of this theorem, in particular regarding transcendence questions.

De nouvelles applications du théorème du sous-espace de Wolfgang Schmidt, certaines assez inattendues, ont été trouvées lors de la dernière décennie. Nous en présentons quelques-unes, en insistant tout particulièrement sur les conséquences des versions quantitatives de ce théorème, notamment concernant des questions de transcendance.

@article{JTNB_2011__23_1_35_0,
     author = {Bugeaud, Yann},
     title = {Quantitative versions of the Subspace Theorem and applications},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     pages = {35-57},
     doi = {10.5802/jtnb.749},
     mrnumber = {2780618},
     zbl = {1272.11089},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2011__23_1_35_0}
}
Bugeaud, Yann. Quantitative versions of the Subspace Theorem and applications. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 35-57. doi : 10.5802/jtnb.749. http://www.numdam.org/item/JTNB_2011__23_1_35_0/

[1] B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers, II. Continued fractions, Acta Math. 195 (2005), 1–20. | MR 2233683 | Zbl 1195.11093

[2] B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in integer bases. Ann. of Math. 165 (2007), 547–565. | MR 2299740 | Zbl 1195.11094

[3] B. Adamczewski and Y. Bugeaud, On the Maillet–Baker continued fractions. J. reine angew. Math. 606 (2007), 105–121. | MR 2337643 | Zbl 1145.11054

[4] B. Adamczewski and Y. Bugeaud, Palindromic continued fractions. Ann. Inst. Fourier (Grenoble) 57 (2007), 1557–1574. | Numdam | MR 2364142 | Zbl 1126.11036

[5] B. Adamczewski et Y. Bugeaud, Mesures de transcendance et aspects quantitatifs de la méthode de Thue–Siegel–Roth–Schmidt. Proc. London Math. Soc. 101 (2010), 1–31. | MR 2661240 | Zbl 1200.11054

[6] B. Adamczewski et Y. Bugeaud, Nombres réels de complexité sous-linéaire : mesures d’irrationalité et de transcendance. J. reine angew. Math. À paraître. | MR 2831513 | Zbl 1255.11037

[7] B. Adamczewski, Y. Bugeaud, and L. Davison, Continued fractions and transcendental numbers. Ann. Inst. Fourier (Grenoble) 56 (2006), 2093–2113. | Numdam | MR 2290775 | Zbl 1152.11034

[8] B. Adamczewski, Y. Bugeaud et F. Luca, Sur la complexité des nombres algébriques. C. R. Acad. Sci. Paris 339 (2004), 11–14. | MR 2075225 | Zbl 1119.11019

[9] P. B. Allen, On the multiplicity of linear recurrence sequences. J. Number Theory 126 (2007), 212–216. | MR 2354929 | Zbl 1133.11006

[10] J.-P. Allouche, Nouveaux résultats de transcendance de réels à développements non aléatoire. Gaz. Math. 84 (2000), 19–34. | MR 1766087

[11] F. Amoroso and E. Viada, Small points on subvarieties of a torus. Duke Math. J. 150 (2009), 407–442. | MR 2582101 | Zbl pre05688238

[12] F. Amoroso and E. Viada, On the zeros of linear recurrence sequences. Preprint.

[13] A. Baker, On Mahler’s classification of transcendental numbers. Acta Math. 111 (1964), 97–120. | MR 157943 | Zbl 0147.03403

[14] Yu. Bilu, The many faces of the subspace theorem [after Adamczewski, Bugeaud, Corvaja, Zannier...]. Séminaire Bourbaki. Vol. 2006/2007. Astérisque No. 317 (2008), Exp. No. 967, vii, 1–38. | MR 2487729 | Zbl pre05382585

[15] E. Bombieri and W. Gubler, Heights in Diophantine geometry. New Mathematical Monographs, vol. 4, Cambridge University Press, 2006. | MR 2216774 | Zbl 1115.11034

[16] Y. Bugeaud, Approximation by algebraic numbers. Cambridge Tracts in Mathematics 160, Cambridge, 2004. | MR 2136100 | Zbl 1055.11002

[17] Y. Bugeaud, Extensions of the Cugiani-Mahler Theorem. Ann. Scuola Normale Superiore di Pisa 6 (2007), 477–498. | Numdam | MR 2370270 | Zbl 1139.11032

[18] Y. Bugeaud, An explicit lower bound for the block complexity of an algebraic number. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), 229–235. | MR 2439519 | Zbl pre05787954

[19] Y. Bugeaud, On the approximation to algebraic numbers by algebraic numbers. Glas. Mat. 44 (2009), 323–331. | MR 2587305 | Zbl 1194.11076

[20] Y. Bugeaud, P. Corvaja, and U. Zannier, An upper bound for the G.C.D. of a n -1 and b n -1. Math. Z. 243 (2003), 79–84. | MR 1953049 | Zbl 1021.11001

[21] Y. Bugeaud and J.-H. Evertse, On two notions of complexity of algebraic numbers. Acta Arith. 133 (2008), 221–250. | MR 2434602 | Zbl pre05309410

[22] Y. Bugeaud and J.-H. Evertse, Approximation of complex algebraic numbers by algebraic numbers of bounded degree. Ann. Scuola Normale Superiore di Pisa 8 (2009), 333–368. | Numdam | MR 2548250 | Zbl 1176.11031

[23] Y. Bugeaud and F. Luca, A quantitative lower bound for the greatest prime factor of (ab+1)(bc+1)(ca+1). Acta Arith. 114 (2004), 275–294. | MR 2071083 | Zbl 1122.11060

[24] P. Bundschuh und A. Pethő, Zur Transzendenz gewisser Reihen. Monatsh. Math. 104 (1987), 199–223. | MR 918473 | Zbl 0601.10025

[25] P. Corvaja and U. Zannier, Diophantine equations with power sums and universal Hilbert sets. Indag. Math. (N.S.) 9 (1998), 317–332. | MR 1692189 | Zbl 0923.11103

[26] P. Corvaja and U. Zannier, Some new applications of the subspace theorem. Compositio Math. 131 (2002), 319–340. | MR 1905026 | Zbl 1010.11038

[27] P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1). Proc. Amer. Math. Soc. 131 (2003), 1705–1709. | MR 1955256 | Zbl 1077.11052

[28] M. Cugiani, Sull’approssimazione di numeri algebrici mediante razionali. Collectanea Mathematica, Pubblicazioni dell’Istituto di matematica dell’Università di Milano 169, Ed. C. Tanburini, Milano, pagg. 5 (1958).

[29] M. Cugiani, Sulla approssimabilità dei numeri algebrici mediante numeri razionali. Ann. Mat. Pura Appl. 48 (1959), 135–145. | MR 112880 | Zbl 0093.05402

[30] M. Cugiani, Sull’approssimabilità di un numero algebrico mediante numeri algebrici di un corpo assegnato. Boll. Un. Mat. Ital. 14 (1959), 151–162. | MR 117220 | Zbl 0086.26402

[31] H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160–167. | MR 77577 | Zbl 0066.29302

[32] E. Dubois et G. Rhin, Approximations rationnelles simultanées de nombres algébriques réels et de nombres algébriques p-adiques. In: Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), pp. 211–227. Astérisque, Nos. 24–25, Soc. Math. France, Paris, 1975. | MR 374042 | Zbl 0305.10031

[33] J.-H. Evertse, On sums of S-units and linear recurrences. Compositio Math. 53 (1984), 225–244. | Numdam | MR 766298 | Zbl 0547.10008

[34] J.-H. Evertse, An explicit version of Faltings’ product theorem and an improvement of Roth’s lemma. Acta Arith. 73 (1995), 215–248. | MR 1364461 | Zbl 0857.11034

[35] J.-H. Evertse, The number of algebraic numbers of given degree approximating a given algebraic number. In: Analytic number theory (Kyoto, 1996), 53–83, London Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, Cambridge, 1997. | MR 1694985 | Zbl 0919.11048

[36] J.-H. Evertse, On the Quantitative Subspace Theorem. Zapiski Nauchnyk Seminarov POMI 377 (2010), 217–240.

[37] J.-H. Evertse and R. G. Ferretti, A further quantitative improvement of the Absolute Subspace Theorem. Preprint.

[38] J.-H. Evertse and K. Győry, Finiteness criteria for decomposable form equations. Acta Arith. 50 (1988), 357–379. | MR 961695 | Zbl 0595.10013

[39] J.-H. Evertse and K. Győry, The number of families of solutions of decomposable form equations. Acta Arith. 80 (1997), 367–394. | MR 1450929 | Zbl 0886.11015

[40] J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, On S-unit equations in two unknowns. Invent. Math. 92 (1988), 461–477. | MR 939471 | Zbl 0662.10012

[41] J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, S-unit equations and their applications. In: New advances in transcendence theory (Durham, 1986), 110–174, Cambridge Univ. Press, Cambridge, 1988. | MR 971998 | Zbl 0658.10023

[42] J.-H. Evertse and H.P. Schlickewei, A quantitative version of the Absolute Subspace Theorem. J. reine angew. Math. 548 (2002), 21–127. | MR 1915209 | Zbl 1026.11060

[43] J.-H. Evertse, H.P. Schlickewei, and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group. Ann. of Math. 155 (2002), 807–836. | MR 1923966 | Zbl 1026.11038

[44] S. Ferenczi and Ch. Mauduit, Transcendence of numbers with a low complexity expansion. J. Number Theory 67 (1997), 146–161. | MR 1486494 | Zbl 0895.11029

[45] K. Győry, Some recent applications of S-unit equations. Journées Arithmétiques, 1991 (Geneva). Astérisque No. 209 (1992), 11, 17–38. | MR 1211001 | Zbl 0792.11005

[46] K. Győry, On the numbers of families of solutions of systems of decomposable form equations. Publ. Math. Debrecen 42 (1993), 65–101. | MR 1208854 | Zbl 0792.11004

[47] K. Győry, On the irreducibility of neighbouring polynomials. Acta Arith. 67 (1994), 283–294. | MR 1292740 | Zbl 0814.11050

[48] K. Győry, A. Sárközy and C. L. Stewart, On the number of prime factors of integers of the form ab+1. Acta Arith. 74 (1996), 365–385. | MR 1378230 | Zbl 0857.11047

[49] S. Hernández and F. Luca, On the largest prime factor of (ab+1)(ac+1)(bc+1). Bol. Soc. Mat. Mexicana 9 (2003), 235–244. | MR 2029272 | Zbl 1108.11030

[50] J. F. Koksma, Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen. Monatsh. Math. Phys. 48 (1939), 176–189. | MR 845 | Zbl 0021.20804

[51] M. Laurent, Équations diophantiennes exponentielles. Invent. Math. 78 (1984), 299–327. | MR 767195 | Zbl 0554.10009

[52] K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II. J. reine angew. Math. 166 (1932), 118–150. | Zbl 0003.38805

[53] K. Mahler, Lectures on Diophantine approximation, Part 1: g-adic numbers and Roth’s theorem. University of Notre Dame, Ann Arbor, 1961. | MR 142509 | Zbl 0158.29903

[54] K. Mahler, Some suggestions for further research. Bull. Austral. Math. Soc. 29 (1984), 101–108. | MR 732177 | Zbl 0517.10001

[55] M. Mignotte, Quelques remarques sur l’approximation rationnelle des nombres algébriques. J. reine angew. Math. 268/269 (1974), 341–347. | MR 357336 | Zbl 0284.10011

[56] M. Mignotte, An application of W. Schmidt’s theorem: transcendental numbers and golden number. Fibonacci Quart. 15 (1977), 15–16. | MR 429781 | Zbl 0353.10025

[57] A. J. van der Poorten and H. P. Schlickewei, The growth condition for recurrence sequences. Macquarie Univ. Math. Rep. 82–0041, North Ryde, Australia (1982).

[58] D. Ridout, Rational approximations to algebraic numbers. Mathematika 4 (1957), 125–131. | MR 93508 | Zbl 0079.27401

[59] K. F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955), 1–20; corrigendum, 168. | MR 72182 | Zbl 0064.28501

[60] H. P. Schlickewei, Die p-adische Verallgemeinerung des Satzes von Thue-Siegel-Roth-Schmidt. J. reine angew. Math. 288 (1976), 86–105. | MR 422166 | Zbl 0333.10018

[61] H. P. Schlickewei, Linearformen mit algebraischen koeffizienten. Manuscripta Math. 18 (1976), 147–185. | MR 401665 | Zbl 0323.10028

[62] H. P. Schlickewei, The 𝔭-adic Thue-Siegel-Roth-Schmidt theorem. Arch. Math. (Basel) 29 (1977), 267–270. | MR 491529 | Zbl 0365.10026

[63] W. M. Schmidt, Über simultane Approximation algebraischer Zahlen durch Rationale. Acta Math. 114 (1965) 159–206. | MR 177948 | Zbl 0136.33802

[64] W. M. Schmidt, On simultaneous approximations of two algebraic numbers by rationals. Acta Math. 119 (1967), 27–50. | MR 223309 | Zbl 0173.04801

[65] W. M. Schmidt, Simultaneous approximations to algebraic numbers by rationals. Acta Math. 125 (1970), 189–201. | MR 268129 | Zbl 0205.06702

[66] W. M. Schmidt, Norm form equations. Ann. of Math. 96 (1972), 526–551. | MR 314761 | Zbl 0226.10024

[67] W. M. Schmidt, Diophantine Approximation. Lecture Notes in Mathematics 785, Springer, 1980. | MR 568710 | Zbl 0421.10019

[68] W. M. Schmidt, The subspace theorem in Diophantine approximation. Compositio Math. 69 (1989), 121–173. | Numdam | MR 984633 | Zbl 0683.10027

[69] W. M. Schmidt, The number of solutions of norm form equations. Trans. Amer. Math. Soc. 317 (1990), 197–227. | MR 961596 | Zbl 0693.10014

[70] W. M. Schmidt, Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics 1467, Springer, 1991. | MR 1176315 | Zbl 0754.11020

[71] W. M. Schmidt, Zeros of linear recurrence sequences. Publ. Math. Debrecen 56 (2000), 609–630. | MR 1766002 | Zbl 0963.11007

[72] Th. Schneider, Über die Approximation algebraischer Zahlen, J. reine angew. Math. 175 (1936), 182–192.

[73] G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers. II. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999), 397–399. | MR 1706560 | Zbl 1003.11035

[74] U. Zannier, Some applications of diophantine approximation to diophantine equations (with special emphasis on the Schmidt subspace theorem). Forum, Udine, 2003.