Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T02:55:30.911Z Has data issue: false hasContentIssue false
  • English
  • Français

ON TRANSCENDENTAL CONTINUED FRACTIONS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  01 October 2021

BÜŞRA CAN Open the ORCID record for BÜŞRA CAN [Opens in a new window]  and
GÜLCAN KEKEÇ Open the ORCID record for GÜLCAN KEKEÇ [Opens in a new window]
BÜŞRA CAN*
Affiliation:
Institute of Graduate Studies in Sciences, Istanbul University, Esnaf Hospital Building, 4th floor, Süleymaniye, Istanbul, Turkey and Department of Maritime Business Management, Faculty of Economics and Administrative Sciences, Piri Reis University, Postane District, Eflatun Street, No. 8, Tuzla 34940, Istanbul, Turkey e-mail: [email protected]
GÜLCAN KEKEÇ
Affiliation:
Department of Mathematics, Faculty of Science, Istanbul University, 34134 Vezneciler, Istanbul, Turkey e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$ -numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$ .

Keywords

Mahler’s classification of transcendental formal power series over a finite fieldU-numbercontinued fractiontranscendence measure

MSC classification

Primary: 11J61: Approximation in non-Archimedean valuations
Secondary: 11J70: Continued fractions and generalizations 11J82: Measures of irrationality and of transcendence
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 392 - 403
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alnıaçık, K., ‘On ${U}_m$ -numbers’, Proc. Amer. Math. Soc. 85 (1982), 499–505.Google Scholar
Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics, 160 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Bundschuh, P., ‘Transzendenzmasse in Körpern formaler Laurentreihen’, J. reine angew. Math. 299–300 (1978), 411–432.Google Scholar
Chaichana, T., Laohakosol, V. and Harnchoowong, A., ‘Linear independence of continued fractions in the field of formal series over a finite field’, Thai J. Math. 4 (2006), 163–177.Google Scholar
İçen, O. Ş., ‘Über die Funktionswerte der $p$ -adisch elliptischen Funktionen I’, İstanb. Üniv. Fen Fak. Mecm. Ser. A 36 (1971), 53–87.Google Scholar
İçen, O. Ş., ‘Anhang zu den Arbeiten “Über die Funktionswerte der $p$ -adisch elliptischen Funktionen I, II”’, İstanb. Üniv. Fen Fak. Mecm. Ser. A 38 (1973), 25–35.Google Scholar
Kekeç, G., ‘ $U$ -numbers in fields of formal power series over finite fields’, Bull. Aust. Math. Soc. 101 (2020), 218–225.CrossRefGoogle Scholar
Kekeç, G., ‘On transcendental formal power series over finite fields’, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 63(111) (2020), 349–357.Google Scholar
Koksma, J. F., ‘Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen’, Monatsh. Math. Phys. 48 (1939), 176–189.CrossRefGoogle Scholar
Mahler, K., ‘Zur Approximation der Exponentialfunktion und des Logarithmus I, II’, J. reine angew. Math. 166 (1932), 118–150.Google Scholar
Müller, R., ‘Algebraische Unabhängigkeit der Werte gewisser Lückenreihen in nicht-archimedisch bewerteten Körpern’, Results Math. 24 (1993), 288–297.CrossRefGoogle Scholar
Ooto, T., ‘ Quadratic approximation in ${F}_q(({T}^{-1}))$ ’, Osaka J. Math. 54 (2017), 129–156.Google Scholar
Oryan, M. H., ‘Über die Unterklassen ${U}_m$ der Mahlerschen Klasseneinteilung der transzendenten formalen Laurentreihen’, İstanb. Üniv. Fen Fak. Mecm. Ser. A 45 (1980), 43–63.Google Scholar
Schmidt, W. M., ‘On continued fractions and diophantine approximation in power series fields’, Acta Arith. 95 (2000), 139–166.CrossRefGoogle Scholar
Schneider, T., Einführung in die Transzendenten Zahlen (Springer, Berlin–Göttingen–Heidelberg, 1957).Google Scholar