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MAHLER’S AND KOKSMA’S CLASSIFICATIONS IN FIELDS OF POWER SERIES

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  07 June 2021

JASON BELL Open the ORCID record for JASON BELL [Opens in a new window]  and
YANN BUGEAUD Open the ORCID record for YANN BUGEAUD [Opens in a new window]
JASON BELL
Affiliation:
Department of Pure Mathematics University of WaterlooWaterloo, ONN2L [email protected]
YANN BUGEAUD*
Affiliation:
Université de Strasbourg Département de mathématiques 7, rue René Descartes, 67084StrasbourgFrance
*
[email protected]
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Abstract

Let q a prime power and ${\mathbb F}_q$ the finite field of q elements. We study the analogues of Mahler’s and Koksma’s classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$ . Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

Keywords

Diophantine approximationpower series fieldMahler’s classification

MSC classification

Primary: 11J61: Approximation in non-Archimedean valuations
Secondary: 11J04: Homogeneous approximation to one number 11J81: Transcendence (general theory)
Type
Article
Information
Nagoya Mathematical Journal , Volume 246 , June 2022 , pp. 355 - 371
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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References

Abhyankar, S., Two notes on formal power series, Proc. Amer. Math. Soc. 7 (1956), 903–905.CrossRefGoogle Scholar
Ayadi, K. and Ooto, T., On quadratic approximation for hyperquadratic continued fractions, J. Number Theory 215 (2020), 171–185.CrossRefGoogle Scholar
Bosch, S., Güntzer, U., and Remmert, R., Non-Archimedean Analysis, Grundlehren der mathematischen Wissenschaften 261, Springer, Berlin, Heidelberg, New York, and Tokyo, 1984.CrossRefGoogle Scholar
Bugeaud, Y., Mahler’s classification of numbers compared with Koksma’s, Acta Arith. 110 (2003), 89–105.CrossRefGoogle Scholar
Bugeaud, Y., Approximation by Algebraic Numbers, Camb. Tracts Math. 160, Cambridge University Press, Cambridge, MA, 2004.CrossRefGoogle Scholar
Bugeaud, Y., Continued fractions with low complexity: Transcendence measures and quadratic approximation, Compos. Math. 148 (2012), 718–750.CrossRefGoogle Scholar
Bugeaud, Y., “Exponents of diophantine approximation” in Dynamics and Analytic Number Theory, London Math. Soc. Lecture Note Ser. 437, Cambridge University Press, Cambridge, MA, 2016, 96–135.CrossRefGoogle Scholar
Bundschuh, P., Transzendenzmaße in Körpern formaler Laurentreihen, J. Reine Angew. Math. 299 (1978), 411–432.Google Scholar
Chen, H.-J., Distribution of Diophantine approximation exponents for algebraic quantities in finite characteristic, J. Number Theory 133 (2013), 3620–3644.CrossRefGoogle Scholar
Chen, H.-J., Baker–Schmidt theorem for Hausdorff dimensions in finite characteristic, Finite Fields Appl. 52 (2018), 336–360.CrossRefGoogle Scholar
Firicel, A., Rational approximations to algebraic Laurent series with coefficients in a finite field, Acta Arith. 157 (2013), 297–322.CrossRefGoogle Scholar
Guntermann, N., Approximation durch algebraische Zahlen beschränkten Grades im Körper der formalen Laurentreihen, Monatsh. Math. 122 (1996), 345–354.CrossRefGoogle Scholar
Kedlaya, K., The algebraic closure of the power series field in positive characteristic, Proc. Amer. Math. Soc. 129 (2001), 3461–3470.CrossRefGoogle Scholar
Kedlaya, K., On the algebraicity of generalized power series, Beitr. Algebra Geom. 58 (2017), 499–527.CrossRefGoogle Scholar
Koksma, J. F., Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monats. Math. Phys. 48 (1939), 176–189.CrossRefGoogle Scholar
Mahler, K., Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. Reine Angew. Math. 166 (1932), 118–150.Google Scholar
Mahler, K., An analogue to Minkowski’s geometry of numbers in a field of series, Ann. of Math. (2) 42 (1941), 488–522.CrossRefGoogle Scholar
Mahler, K., On a theorem of Liouville in fields of positive characteristic, Canad. J. Math. 1 (1949), 397–400.CrossRefGoogle Scholar
Ooto, T., Quadratic approximation in Fq((T −1)), Osaka J. Math. 54 (2017), 129–156.Google Scholar
Ooto, T., On Diophantine exponents for Laurent series over a finite field, J. Number Theory 185 (2018), 349–378.CrossRefGoogle Scholar
Ooto, T., The existence of T-numbers in positive characteristic, Acta Arith. 189 (2019), 179–189.CrossRefGoogle Scholar
Schmidt, W. M., “T-numbers do exist” in Symposia Mathematica, Vol. IV (Inst. Naz. di Alta Math., Rome, 1968), Academic Press, London, 1970, 3–26.Google Scholar
Schmidt, W. M., “Mahler’s T-numbers” in 1969 Number Theory Institute (Proceedings of Symposia in Pure Mathematics, Vol. 20, State University of New York, Stony Brook, NY, 1969), Amer. Math. Soc., Providence, RI, 1971, 275–286.CrossRefGoogle Scholar
Sprindžuk, V. G., Mahler’s Problem in Metric Number Theory, Amer. Math. Soc., Providence, RI, 1969.Google Scholar
Thakur, D. S., Higher Diophantine approximation exponents and continued fraction symmetries for function fields, Proc. Amer. Math. Soc. 139 (2011), 11–19.CrossRefGoogle Scholar
Thakur, D. S., “From rationality to transcendence in finite characteristic” in Transcendance et irrationalité, SMF Journ. Annu. 2012, Soc. Math.France, Montrouge, 2012, 21–48.Google Scholar
Thakur, D. S., Higher Diophantine approximation exponents and continued fraction symmetries for function fields II, Proc. Amer. Math. Soc. 141 (2013), 2603–2608.CrossRefGoogle Scholar
Wirsing, E., Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. 206 (1961), 67–77.CrossRefGoogle Scholar
Wirsing, E., “On approximations of algebraic numbers by algebraic numbers of bounded degree” in 1969 Number Theory Institute (Proceedings of Symposia in Pure Mathematics, Vol. 20, State University of New York, Stony Brook, NY, 1969), Amer. Math. Soc., Providence, RI, 1971, 213–247.CrossRefGoogle Scholar