On a theorem of Nathanson on Diophantine approximation
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Jaroslav Hančl
und
Tho Phuoc Nguyen
Abstract
In 1974, M. B. Nathanson proved that every irrational number α represented by a simple continued fraction with infinitely many elements greater than or equal to k, is approximable by an infinite number of rational numbers p/q satisfying
Acknowledgement
The authors thank unknown referee for valuable suggestions.
(Communicated by Milan Paštéka)
References
[1] Bahnerová, S.—Hančl, J.: Sharpening of the theorem of Vahlen and related theorems, J. Ramanujan. Math. Soc. 36(2) (2021), 109–121.Suche in Google Scholar
[2] Borel, É.: Sur l’approximation des nombres par des nombres rationnels, C. R. Acad. Sci. Paris 136 (1903), 1054–1055.Suche in Google Scholar
[3] Borel, É.: Contribution `a l’analyse arithmétique du continu, J. Math. Pures 9(5) (1903), 329–375.Suche in Google Scholar
[4] Borwein, J.—Borwein, P.: Pi and the Agm: A Study in Analytic Number Theory and Computational Complexity, John Wiley & Sons, New York, 1987.Suche in Google Scholar
[5] Cohn, J. H. E.: Hurwitz’s theorem, Proc. Amer. Math. Soc. 38 (1973), 436.10.1090/S0002-9939-1973-0313195-9Suche in Google Scholar
[6] Dirichlet, L. G. P.: Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einige Anwendungen auf die Theorie der Zahlen, S.-B. Preuss. Akad. Wiss. (1842), 93–95.Suche in Google Scholar
[7] Fel’Dman, N. I.—Nesterenko, Y. V.: Transcendental Numbers. Encyclopaedia of Mathematical Sciences, Vol. 44: Number Theory IV (A. N. Parshin and I. R. Shafarevich, eds.), Springer-Verlag, New York, 1998.Suche in Google Scholar
[8] Hančl, J.: Sharpening of theorems of Vahlen and Hurwitz and approximation properties of the golden ratio, Arch. Math. (Basel) 105(2) (2015), 129–137.10.1007/s00013-015-0788-8Suche in Google Scholar
[9] Hančl, J.: Second basic theorem of Hurwitz, Lith. Math. J. 56(1) (2016), 72–76.10.1007/s10986-016-9305-4Suche in Google Scholar
[10] Hančl, J.—Nair, R.: On a theorem of Borel on Diophantine approximation, Ramanujan J. (2024); https://doi.org/10.1007/s11139-024-00922-6.10.1007/s11139-024-00922-6Suche in Google Scholar
[11] Hardy, G. H.—Wright, E. M.: An Introduction to the Theory of Numbers, 6th ed., revised by D. R. Heath-Brown and J. H. Silverman with a foreword by Andrew Wiles, Oxford University Press, Oxford, 2008.10.1093/oso/9780199219858.002.0002Suche in Google Scholar
[12] Hensley, D.: Continued Fractions, Word Scientific Publishing, 2006.10.1142/9789812774682Suche in Google Scholar
[13] Hurwitz, A.: Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Math. Ann. 39(2) (1891), 279–284 (in German).10.1007/BF01206656Suche in Google Scholar
[14] Jones, W. B.—Thron, W. J.: Continued Fractions Analytic Theory and Applications, Encyclopedia of Mathematics and its applications, Vol. 11, Cambridge University Press, 1984.10.1017/CBO9780511759550Suche in Google Scholar
[15] Karpenkov, O.: Geometry of Continued Fractions, Algorithms and Computation in Mathematics, Vol. 26, Springer, Heidelberg, 2013.10.1007/978-3-642-39368-6Suche in Google Scholar
[16] Khinchin, A. Y.: Continued fractions, The University of Chicago Press, Chicago, (1964).10.1063/1.3051235Suche in Google Scholar
[17] Nathanson M. B.: Approximation by continued fractions, Proc Amer. Math. Soc. 45 (1974), 323–324.10.1090/S0002-9939-1974-0349594-XSuche in Google Scholar
[18] Rosen, K. H.: Elementary Number Theory and its Applications, Addison Wesley, fifth edition, 2005.Suche in Google Scholar
[19] Schmidt, W.: Diophantine Approximation. Lecture Notes in Math., Vol. 785, Springer, Berlin, 1980.Suche in Google Scholar
[20] Vahlen, K. TH.: Über Näherungswerte und Kettenbrüche, J. Reine Angew. Math. 115 (1895), 221–233.10.1515/crll.1895.115.221Suche in Google Scholar
[21] Wall, H. S.: Analytic Theory of Continued Fractions, New York, Chelsea, 1948.Suche in Google Scholar
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Artikel in diesem Heft
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- On a theorem of Nathanson on Diophantine approximation
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- Advancing reliability and medical data analysis through novel statistical distribution exploration
- Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
Artikel in diesem Heft
- Generalized Sasaki mappings in d0-Algebras
- On a theorem of Nathanson on Diophantine approximation
- Constructing infinite families of number fields with given indices from quintinomials
- Partitions into two Lehmer numbers in ℤq
- Fundamental systems of solutions of some linear differential equations of higher order
- On k-Circulant matrices involving the Lucas numbers of even index
- Explicit formulae for the Drazin inverse of the sum of two matrices
- On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
- Novel generalized tempered fractional integral inequalities for convexity property and applications
- Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
- The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
- Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
- Improvements on the Leighton oscillation theorem for second-order dynamic equations
- Topogenous orders on forms
- Comparison of topologies on fundamental groups with subgroup topology viewpoint
- An elementary proof of the generalized Itô formula
- Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
- Advancing reliability and medical data analysis through novel statistical distribution exploration
- Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
