Quantitative versions of the Subspace Theorem and applications
Yann Bugeaud[1]
- [1] Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg Cedex (France)
Journal de Théorie des Nombres de Bordeaux (2011)
- Volume: 23, Issue: 1, page 35-57
- ISSN: 1246-7405
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topBugeaud, Yann. "Quantitative versions of the Subspace Theorem and applications." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 35-57. <http://eudml.org/doc/219672>.
@article{Bugeaud2011,
abstract = {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.},
affiliation = {Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg Cedex (France)},
author = {Bugeaud, Yann},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Roth theorem; subspace theorem; quantitative subspace theorem; approximation of algebraic numbers by algebraic numbers; norm form equations; unit equations},
language = {eng},
month = {3},
number = {1},
pages = {35-57},
publisher = {Société Arithmétique de Bordeaux},
title = {Quantitative versions of the Subspace Theorem and applications},
url = {http://eudml.org/doc/219672},
volume = {23},
year = {2011},
}
TY - JOUR
AU - Bugeaud, Yann
TI - Quantitative versions of the Subspace Theorem and applications
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 35
EP - 57
AB - 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.
LA - eng
KW - Roth theorem; subspace theorem; quantitative subspace theorem; approximation of algebraic numbers by algebraic numbers; norm form equations; unit equations
UR - http://eudml.org/doc/219672
ER -
References
top- B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers, II. Continued fractions, Acta Math. 195 (2005), 1–20. Zbl1195.11093MR2233683
- B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in integer bases. Ann. of Math. 165 (2007), 547–565. Zbl1195.11094MR2299740
- B. Adamczewski and Y. Bugeaud, On the Maillet–Baker continued fractions. J. reine angew. Math. 606 (2007), 105–121. Zbl1145.11054MR2337643
- B. Adamczewski and Y. Bugeaud, Palindromic continued fractions. Ann. Inst. Fourier (Grenoble) 57 (2007), 1557–1574. Zbl1126.11036MR2364142
- 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. Zbl1200.11054MR2661240
- B. Adamczewski et Y. Bugeaud, Nombres réels de complexité sous-linéaire : mesures d’irrationalité et de transcendance. J. reine angew. Math. À paraître.
- B. Adamczewski, Y. Bugeaud, and L. Davison, Continued fractions and transcendental numbers. Ann. Inst. Fourier (Grenoble) 56 (2006), 2093–2113. Zbl1152.11034MR2290775
- B. Adamczewski, Y. Bugeaud et F. Luca, Sur la complexité des nombres algébriques. C. R. Acad. Sci. Paris 339 (2004), 11–14. Zbl1119.11019MR2075225
- P. B. Allen, On the multiplicity of linear recurrence sequences. J. Number Theory 126 (2007), 212–216. Zbl1133.11006MR2354929
- J.-P. Allouche, Nouveaux résultats de transcendance de réels à développements non aléatoire. Gaz. Math. 84 (2000), 19–34. MR1766087
- F. Amoroso and E. Viada, Small points on subvarieties of a torus. Duke Math. J. 150 (2009), 407–442. Zbl1234.11081MR2582101
- F. Amoroso and E. Viada, On the zeros of linear recurrence sequences. Preprint. Zbl1271.11015
- A. Baker, On Mahler’s classification of transcendental numbers. Acta Math. 111 (1964), 97–120. Zbl0147.03403MR157943
- 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. Zbl1220.11091MR2487729
- E. Bombieri and W. Gubler, Heights in Diophantine geometry. New Mathematical Monographs, vol. 4, Cambridge University Press, 2006. Zbl1115.11034MR2216774
- Y. Bugeaud, Approximation by algebraic numbers. Cambridge Tracts in Mathematics 160, Cambridge, 2004. Zbl1055.11002MR2136100
- Y. Bugeaud, Extensions of the Cugiani-Mahler Theorem. Ann. Scuola Normale Superiore di Pisa 6 (2007), 477–498. Zbl1139.11032MR2370270
- 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. Zbl1268.11099MR2439519
- Y. Bugeaud, On the approximation to algebraic numbers by algebraic numbers. Glas. Mat. 44 (2009), 323–331. Zbl1194.11076MR2587305
- Y. Bugeaud, P. Corvaja, and U. Zannier, An upper bound for the G.C.D. of and . Math. Z. 243 (2003), 79–84. Zbl1021.11001MR1953049
- Y. Bugeaud and J.-H. Evertse, On two notions of complexity of algebraic numbers. Acta Arith. 133 (2008), 221–250. Zbl1236.11062MR2434602
- 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. Zbl1176.11031MR2548250
- Y. Bugeaud and F. Luca, A quantitative lower bound for the greatest prime factor of . Acta Arith. 114 (2004), 275–294. Zbl1122.11060MR2071083
- P. Bundschuh und A. Pethő, Zur Transzendenz gewisser Reihen. Monatsh. Math. 104 (1987), 199–223. Zbl0601.10025MR918473
- P. Corvaja and U. Zannier, Diophantine equations with power sums and universal Hilbert sets. Indag. Math. (N.S.) 9 (1998), 317–332. Zbl0923.11103MR1692189
- P. Corvaja and U. Zannier, Some new applications of the subspace theorem. Compositio Math. 131 (2002), 319–340. Zbl1010.11038MR1905026
- P. Corvaja and U. Zannier, On the greatest prime factor of . Proc. Amer. Math. Soc. 131 (2003), 1705–1709. Zbl1077.11052MR1955256
- 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).
- M. Cugiani, Sulla approssimabilità dei numeri algebrici mediante numeri razionali. Ann. Mat. Pura Appl. 48 (1959), 135–145. Zbl0093.05402MR112880
- M. Cugiani, Sull’approssimabilità di un numero algebrico mediante numeri algebrici di un corpo assegnato. Boll. Un. Mat. Ital. 14 (1959), 151–162. Zbl0086.26402MR117220
- H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160–167. Zbl0066.29302MR77577
- E. Dubois et G. Rhin, Approximations rationnelles simultanées de nombres algébriques réels et de nombres algébriques -adiques. In: Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), pp. 211–227. Astérisque, Nos. 24–25, Soc. Math. France, Paris, 1975. Zbl0305.10031MR374042
- J.-H. Evertse, On sums of -units and linear recurrences. Compositio Math. 53 (1984), 225–244. Zbl0547.10008MR766298
- J.-H. Evertse, An explicit version of Faltings’ product theorem and an improvement of Roth’s lemma. Acta Arith. 73 (1995), 215–248. Zbl0857.11034MR1364461
- 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. Zbl0919.11048MR1694985
- J.-H. Evertse, On the Quantitative Subspace Theorem. Zapiski Nauchnyk Seminarov POMI 377 (2010), 217–240. Zbl1282.11085
- J.-H. Evertse and R. G. Ferretti, A further quantitative improvement of the Absolute Subspace Theorem. Preprint. Zbl06156615
- J.-H. Evertse and K. Győry, Finiteness criteria for decomposable form equations. Acta Arith. 50 (1988), 357–379. Zbl0595.10013MR961695
- J.-H. Evertse and K. Győry, The number of families of solutions of decomposable form equations. Acta Arith. 80 (1997), 367–394. Zbl0886.11015MR1450929
- J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, On -unit equations in two unknowns. Invent. Math. 92 (1988), 461–477. Zbl0662.10012MR939471
- J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, -unit equations and their applications. In: New advances in transcendence theory (Durham, 1986), 110–174, Cambridge Univ. Press, Cambridge, 1988. Zbl0658.10023MR971998
- J.-H. Evertse and H.P. Schlickewei, A quantitative version of the Absolute Subspace Theorem. J. reine angew. Math. 548 (2002), 21–127. Zbl1026.11060MR1915209
- 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. Zbl1026.11038MR1923966
- S. Ferenczi and Ch. Mauduit, Transcendence of numbers with a low complexity expansion. J. Number Theory 67 (1997), 146–161. Zbl0895.11029MR1486494
- K. Győry, Some recent applications of -unit equations. Journées Arithmétiques, 1991 (Geneva). Astérisque No. 209 (1992), 11, 17–38. Zbl0792.11005MR1211001
- K. Győry, On the numbers of families of solutions of systems of decomposable form equations. Publ. Math. Debrecen 42 (1993), 65–101. Zbl0792.11004MR1208854
- K. Győry, On the irreducibility of neighbouring polynomials. Acta Arith. 67 (1994), 283–294. Zbl0814.11050MR1292740
- K. Győry, A. Sárközy and C. L. Stewart, On the number of prime factors of integers of the form . Acta Arith. 74 (1996), 365–385. Zbl0857.11047MR1378230
- S. Hernández and F. Luca, On the largest prime factor of . Bol. Soc. Mat. Mexicana 9 (2003), 235–244. Zbl1108.11030MR2029272
- 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. Zbl0021.20804MR845
- M. Laurent, Équations diophantiennes exponentielles. Invent. Math. 78 (1984), 299–327. Zbl0554.10009MR767195
- K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II. J. reine angew. Math. 166 (1932), 118–150. Zbl0003.38805
- K. Mahler, Lectures on Diophantine approximation, Part 1: -adic numbers and Roth’s theorem. University of Notre Dame, Ann Arbor, 1961. Zbl0158.29903MR142509
- K. Mahler, Some suggestions for further research. Bull. Austral. Math. Soc. 29 (1984), 101–108. Zbl0517.10001MR732177
- M. Mignotte, Quelques remarques sur l’approximation rationnelle des nombres algébriques. J. reine angew. Math. 268/269 (1974), 341–347. Zbl0284.10011MR357336
- M. Mignotte, An application of W. Schmidt’s theorem: transcendental numbers and golden number. Fibonacci Quart. 15 (1977), 15–16. Zbl0353.10025MR429781
- 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).
- D. Ridout, Rational approximations to algebraic numbers. Mathematika 4 (1957), 125–131. Zbl0079.27401MR93508
- K. F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955), 1–20; corrigendum, 168. Zbl0064.28501MR72182
- H. P. Schlickewei, Die -adische Verallgemeinerung des Satzes von Thue-Siegel-Roth-Schmidt. J. reine angew. Math. 288 (1976), 86–105. Zbl0333.10018MR422166
- H. P. Schlickewei, Linearformen mit algebraischen koeffizienten. Manuscripta Math. 18 (1976), 147–185. Zbl0323.10028MR401665
- H. P. Schlickewei, The -adic Thue-Siegel-Roth-Schmidt theorem. Arch. Math. (Basel) 29 (1977), 267–270. Zbl0365.10026MR491529
- W. M. Schmidt, Über simultane Approximation algebraischer Zahlen durch Rationale. Acta Math. 114 (1965) 159–206. Zbl0136.33802MR177948
- W. M. Schmidt, On simultaneous approximations of two algebraic numbers by rationals. Acta Math. 119 (1967), 27–50. Zbl0173.04801MR223309
- W. M. Schmidt, Simultaneous approximations to algebraic numbers by rationals. Acta Math. 125 (1970), 189–201. Zbl0205.06702MR268129
- W. M. Schmidt, Norm form equations. Ann. of Math. 96 (1972), 526–551. Zbl0226.10024MR314761
- W. M. Schmidt, Diophantine Approximation. Lecture Notes in Mathematics 785, Springer, 1980. Zbl0421.10019MR568710
- W. M. Schmidt, The subspace theorem in Diophantine approximation. Compositio Math. 69 (1989), 121–173. Zbl0683.10027MR984633
- W. M. Schmidt, The number of solutions of norm form equations. Trans. Amer. Math. Soc. 317 (1990), 197–227. Zbl0693.10014MR961596
- W. M. Schmidt, Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics 1467, Springer, 1991. Zbl0754.11020MR1176315
- W. M. Schmidt, Zeros of linear recurrence sequences. Publ. Math. Debrecen 56 (2000), 609–630. Zbl0963.11007MR1766002
- Th. Schneider, Über die Approximation algebraischer Zahlen, J. reine angew. Math. 175 (1936), 182–192.
- 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. Zbl1003.11035MR1706560
- U. Zannier, Some applications of diophantine approximation to diophantine equations (with special emphasis on the Schmidt subspace theorem). Forum, Udine, 2003.
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