Quantitative versions of the Subspace Theorem and applications

Bugeaud, Yann

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 35-57.

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.

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.

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[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. | Numdam | MR

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

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

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

[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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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. | Numdam | MR | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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