Doc. Math. Extra Vol. Mahler Selecta, 79-93 (2019)
DOI: 10.25537/dm.2019.SB-79-93
Mathematics Subject Classification
11-03, 11J72
References
[M9]. K. Mahler, Über Beziehungen zwischen der Zahl \(e\) und Liouvilleschen Zahlen, Math. Z. 31 (1930), 729-732.
[M11]. K. Mahler, Zur Approximation der Exponentialfunktion und des Logarithmus, I, II, J. Reine Angew. Math. (Crelle) 166 (1932), 118-150. ([M11] and [M13] combined).
[M12]. K. Mahler, Über das Maßder Menge aller \(S\)-Zahlen, Math. Ann. 106 (1932), 131-139.
[M27]. K. Mahler, Über eine Klassen-Einteilung der \(p\)-adischen Zahlen, Mathematica (Zutphen) 3 (1935), 177-185.
[M179]. K. Mahler, On the order function of a transcendental number, Acta Arithm. 18 (1971), 63-76.
1. B. Adamczewski and Y. Bugeaud, Nombres réels de complexité sous-linéaire: mesures d'irrationalité et de transcendance, J. Reine Angew. Math. (Crelle) 658 (2011), 65-98.
2. J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge University Press, Cambridge, 2003.
3. F. Amoroso, On the distribution of complex numbers according to their transcendence types, Ann. Mat. Pura Appl. (4) 151 (1988), 359-368.
4. M. Amou, An improvement of a transcendence measure of Galochkin and Mahler's \(S\)-numbers, J. Aust. Math. Soc. Ser. A 52 (1992), no. 1, 130-140.
5. M. Amou, On Sprindžuk's classification of transcendental numbers, J. Reine Angew. Math. (Crelle) 470 (1996), 27-50.
6. M. Amou and Y. Bugeaud, On integer polynomials with multiple roots, Mathematika 54 (2007), no. 1-2, 83-92.
7. A. Baker, On a theorem of Sprindžuk, Proc. Roy. Soc. London Ser. A 292 (1966), 92-104.
8. A. Baker, Transcendental Number Theory, Cambridge University Press, London, 1975.
9. A. Baker and W. M. Schmidt, Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc. (3) 21 (1970), 1-11.
10. R. C. Baker, On approximation with algebraic numbers of bounded degree, Mathematika 23 (1976), no. 1, 18-31.
11. P.-G. Becker, \(k\)-regular power series and Mahler-type functional equations, J. Number Theory 49 (1994), no. 3, 269-286.
12. J. P. Bell, Y. Bugeaud, and M. Coons, Diophantine approximation of Mahler numbers, Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1157-1206.
13. V. Beresnevich, On approximation of real numbers by real algebraic numbers, Acta Arith. 90 (1999), no. 2, 97-112.
14. V. I. Bernik, The exact order of approximating zero by values of integral polynomials, Acta Arith. 53 (1989), no. 1, 17-28.
15. V. I. Bernik and M. M. Dodson, Metric Diophantine Approximation on Manifolds, Cambridge University Press, Cambridge, 1999.
16. Y. Bugeaud, Approximation by algebraic numbers, Cambridge University Press, Cambridge, 2004.
17. Y. Bugeaud, Mahler's classification of numbers compared with Koksma's. III, Publ. Math. Debrecen 65 (2004), no. 3-4, 305-316.
18. Y. Bugeaud, Continued fractions with low complexity: transcendence measures and quadratic approximation, Compos. Math. 148 (2012), no. 3, 718-750.
19. Y. Bugeaud and A. Dujella, Root separation for irreducible integer polynomials, Bull. Lond. Math. Soc. 43 (2011), no. 6, 1239-1244.
20. G. Chudnovsky, Algebraic independence of the values of elliptic function at algebraic points, Invent. Math. 61 (1980), no. 3, 267-290.
21. A. Durand, Quatre problèmes de Mahler sur la fonction ordre d'un nombre transcendant, Bull. Soc. Math. France 102 (1974), 365-377.
22. A. I. Galočkin, A transcendence measure for the values of functions satisfying certain functional equations, Mat. Zametki 27 (1980), no. 2, 175-183.
23. D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148 (1998), no. 1, 339-360.
24. 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.
25. S. Lang, A transcendence measure for \(E\)-functions, Mathematika 9 (1962), 157-161.
26. S. Lang, Introduction to Transcendental Numbers, Addison-Wesley, Reading, MA, 1966.
27. W. J. LeVeque, On Mahler's \(U\)-numbers, J. London Math. Soc. 28 (1953), 220-229.
28. J. Liouville, Remarques relatives à des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques, C. R. Acad. Sci. Paris 18 (1844), 883-885.
29. Ju. V. Nesterenko, An order function for almost all numbers, Mat. Zametki 15 (1974), 405-414.
30. Ku. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Mathematics, vol. 1631, Springer, Berlin, 1996.
31. T. Pejković, On \(p\)-adic \(T\)-numbers, Publ. Math. Debrecen 82 (2013), no. 3-4, 549-567.
32. P. Philippon, Classification de Mahler et distances locales, Bull. Aust. Math. Soc. 49 (1994), no. 2, 219-238.
33. J. Popken, Zur Transzendenz von \(e\), Math. Z. 29 (1929), no. 1, 525-541.
34. É. Reyssat, Approximation algébrique de nombres liés aux fonctions elliptiques et exponentielle, Bull. Soc. Math. France 108 (1980), no. 1, 47-79.
35. H. P. Schlickewei, \(p\)-adic \(T\)-numbers do exist, Acta Arith. 39 (1981), no. 2, 181-191.
36. W. M. Schmidt, \(T\)-numbers do exist, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 3-26.
37. W. M. Schmidt, Mahler's \(T\)-numbers, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, D. J. Lewis (ed.), State Univ. New York, Stony Brook, NY, 1969), Amer. Math. Soc., Providence, RI, 1971, pp. 275-286.
38. T. Schneider, Einführung in die transzendenten Zahlen, Springer, Berlin, 1957.
39. A. B. Shidlovskii, Transcendental Numbers, de Gruyter, Berlin, 1989.
40. C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuß. Akad. Wiss. Phys.-Math. Kl. 1 (1929), 1-70.
41. V. G. Sprindžuk, On a classification of transcendental numbers, Litovsk. Mat. Sb. 2 (1962), no. 2, 215-219.
42. V. G. Sprindžuk, A proof of Mahler's conjecture on the measure of the set of \(S\)-numbers, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 379-436.
43. V. G. Sprindžuk, Mahler's Problem in Metric Number Theory, American Mathematical Society, Providence, RI, 1969.
44. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation, Marcel Dekker, New York, 1974.
45. E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. (Crelle) 206 (1961), 67-77.
Affiliation
Amou, Masaaki
Department of Mathematics, Gunma University, Tenjin-cho 1-5-1, Kiryu 376-8515, Japan
Bugeaud, Yann
Université de Strasbourg et C.N.R.S., IRMA, U.M.R. 7501, 7 rue René Descartes, 67084 Strasbourg Cedex, France
