On Transcendental Regular Continued Fractions
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Gülcan Kekeç
ABSTRACT
We establish explicit constructions of real transcendental numbers that are not U-numbers with respect to Mahler’s classification by using regular continued fraction expansions of irrational real algebraic numbers.
Funding statement: This research is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) with project number 121F219.
Acknowledgement
The author acknowledges the referees for their valuable suggestions and comments.
REFERENCES
[1] Alniaçik, K.: On the subclasses Um in Mahler’s classification of the transcendental numbers, İstanbul Üniv. Fen Fak. Mecm. Ser. A 44 (1979), 39–82.Search in Google Scholar
[2] Alniaçik, K.: On Um-numbers, Proc. Amer. Math. Soc. 85 (1982), 499–505.10.1090/S0002-9939-1982-0660590-XSearch in Google Scholar
[3] Baker, A.: On Mahler’s classification of transcendental numbers, Acta Math. 111 (1964), 97–120.10.1007/BF02391010Search in Google Scholar
[4] Bell, J.—Bugeaud, Y.: Mahler’s and Koksma’s classifications in fields of power series, Nagoya Math. J. 246 (2022), 355–371.10.1017/nmj.2021.5Search in Google Scholar
[5] Bugeaud, Y.: Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics 160, Cambridge University Press, Cambridge, 2004.10.1017/CBO9780511542886Search in Google Scholar
[6] Bugeaud, Y.—Kekeç, G.: On Mahler’s classification of p-adic numbers, Bull. Aust. Math. Soc. 98 (2018), 203–211.10.1017/S0004972718000515Search in Google Scholar
[7] Bugeaud, Y.—Kekeç, G.: On Mahler’s p-adic S-, T-, and U-numbers, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 28 (2020), 81–94.10.2478/auom-2020-0005Search in Google Scholar
[8] Bundschuh, P.: Transzendenzmasse in Körpern formaler Laurentreihen, J. Reine Angew. Math. 299/300 (1978), 411–432.10.1515/crll.1978.299-300.411Search in Google Scholar
[9] Can, B.—Kekeç, G.: On transcendental continued fractions in fields of formal power series over finite fields, Bull. Aust. Math. Soc. 105 (2022), 392–403.10.1017/S0004972721000800Search in Google Scholar
[10] Kekeç, G.: On Mahler’s S-numbers, T-numbers, and U-numbers, Hacet. J. Math. Stat. 49 (2020), 45–55.10.2307/j.ctv15d7zkw.38Search in Google Scholar
[11] Kekeç, G.: U-numbers in fields of formal power series over finite fields, Bull. Aust. Math. Soc. 101 (2020), 218–225.10.1017/S0004972719000832Search in Google Scholar
[12] Kekeç, G.: On transcendental formal power series over finite fields, Bull. Math. Soc. Sci. Math. Roumanie 63(111) (2020), 349–357.Search in Google Scholar
[13] Kekeç, G.: On Mahler’s classification of formal power series over a finite field, Math. Slovaca 72 (2022), 265–273.10.1515/ms-2022-0017Search in Google Scholar
[14] Leveque, W. J.: On Mahler’s U-numbers, J. London Math. Soc. 28 (1953), 220–229.10.1112/jlms/s1-28.2.220Search in Google Scholar
[15] Leveque, W. J.: Topics in Number Theory Volume II, Addison-Wesley Publishing, 1956.Search in Google Scholar
[16] Mahler, K.: Zur Approximation der Exponentialfunktion und des Logarithmus I, II, J. Reine Angew. Math. 166 (1932), 118–150.10.1515/crll.1932.166.137Search in Google Scholar
[17] Mahler, K.: Über eine Klasseneinteilung der p-adischen Zahlen, Mathematica (Leiden) 3 (1935), 177–185.Search in Google Scholar
[18] Ooto, T.: The existence of T-numbers in positive characteristic, Acta Arith. 189 (2019), 179–189.10.4064/aa180325-24-8Search in Google Scholar
[19] Oryan, M. H.: Über die Unterklassen Um der Mahlerschen Klasseneinteilung der transzendenten formalen Laurentreihen, İstanbul Üniv. Fen Fak. Mecm. Ser. A 45 (1980), 43–63.Search in Google Scholar
[20] Oryan, M. H.: On power series and Mahler’s U-numbers, Math. Scand. 65 (1989), 143–151.10.7146/math.scand.a-12273Search in Google Scholar
[21] Oryan, M. H.: On power series and Mahler’s U-numbers, İstanbul Üniv. Fen Fak. Mecm. Ser. A 47 (1990), 117–125.Search in Google Scholar
[22] Roth, K. F.: Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20; corrigendum, 168.10.1112/S0025579300000644Search in Google Scholar
[23] Schlickewei, H. P.: p-adic T-numbers do exist, Acta Arith. 39 (1981), 181–191.10.4064/aa-39-2-181-191Search in Google Scholar
[24] Schmidt, W. M.: T-numbers do exist. In: Symposia Mathematica Vol. IV (INDAM, Rome, 1968/1969), Academic Press, London, 1970, 3–26.Search in Google Scholar
[25] Sprindžuk, V. G.: Mahler’s Problem in Metric Number Theory. Transl. Math. Monogr. 25, American Mathematical Society, Providence, R.I., 1969.Search in Google Scholar
[26] Zeren, B. M.: Über eine Klasse von verallgemeinerten Lückenreihen, deren Werte für algebraische Argumente transzendent, aber keine U-Zahlen sind I, İstanbul Üniv. Fen Fak. Mat. Derg. 50 (1991), 79–99.Search in Google Scholar
[27] Zeren, B. M.: Über eine Klasse von verallgemeinerten Lückenreihen, deren Werte für algebraische Argumente transzendent, aber keine U-Zahlen sind III, İstanbul Üniv. Fen Fak. Mat. Derg. 50 (1991), 147–158.Search in Google Scholar
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Articles in the same Issue
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- Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications
- Uniform Local Connectedness and Completion of Metric σ-Frames
- On Transcendental Regular Continued Fractions
- On Repdigits Which are Sums or Differences of Two k-Pell Numbers
- Peirce Decompositions for Evolution Algebras
- Generalized Anti-Symmetry Laws in Groupoids
- New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
- Geometric Properties of Generalized Bessel Function Associated with the Exponential Function
- On the Attainable Set of Iterative Differential Inclusions
- On the Differential-Difference Sine-Gordon Equation with an Integral Type Source
- A Critical p(x)-Laplacian Steklov Type Problem with Weights
- Distributivity of a Uni-nullnorm with Continuous and Archimedean Underlying T-norms and T-conorms Over an Arbitrary Uninorm
- Some Approximation Properties of Operators Including Degenerate Appell Polynomials
- Operators that are Weakly Coercive and a Compact Perturbation
- On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric
- Paracompactness and Open Relations
- On (Strongly) (Θ-)Discrete Homogeneous Spaces
- The Alpha Power Muth-G Distributions and Its Applications in Survival and Reliability Analyses
- Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds
Articles in the same Issue
-
- Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications
- Uniform Local Connectedness and Completion of Metric σ-Frames
- On Transcendental Regular Continued Fractions
- On Repdigits Which are Sums or Differences of Two k-Pell Numbers
- Peirce Decompositions for Evolution Algebras
- Generalized Anti-Symmetry Laws in Groupoids
- New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
- Geometric Properties of Generalized Bessel Function Associated with the Exponential Function
- On the Attainable Set of Iterative Differential Inclusions
- On the Differential-Difference Sine-Gordon Equation with an Integral Type Source
- A Critical p(x)-Laplacian Steklov Type Problem with Weights
- Distributivity of a Uni-nullnorm with Continuous and Archimedean Underlying T-norms and T-conorms Over an Arbitrary Uninorm
- Some Approximation Properties of Operators Including Degenerate Appell Polynomials
- Operators that are Weakly Coercive and a Compact Perturbation
- On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric
- Paracompactness and Open Relations
- On (Strongly) (Θ-)Discrete Homogeneous Spaces
- The Alpha Power Muth-G Distributions and Its Applications in Survival and Reliability Analyses
- Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds
