Introduction to Diophantine Approximation

Yasushige Watase

Formalized Mathematics (2015)

  • Volume: 23, Issue: 2, page 101-106
  • ISSN: 1426-2630

Abstract

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In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

How to cite

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Yasushige Watase. "Introduction to Diophantine Approximation." Formalized Mathematics 23.2 (2015): 101-106. <http://eudml.org/doc/271783>.

@article{YasushigeWatase2015,
abstract = {In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].},
author = {Yasushige Watase},
journal = {Formalized Mathematics},
keywords = {irrational number; approximation; continued fraction; rational number; Dirichlet’s proof; Dirichlet's proof},
language = {eng},
number = {2},
pages = {101-106},
title = {Introduction to Diophantine Approximation},
url = {http://eudml.org/doc/271783},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Yasushige Watase
TI - Introduction to Diophantine Approximation
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 2
SP - 101
EP - 106
AB - In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].
LA - eng
KW - irrational number; approximation; continued fraction; rational number; Dirichlet’s proof; Dirichlet's proof
UR - http://eudml.org/doc/271783
ER -

References

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  1. [1] Alan Baker. A Concise Introduction to the Theory of Numbers. Cambridge University Press, 1984. Zbl0554.10001
  2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. 
  3. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
  4. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  5. [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  6. [6] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990. 
  7. [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990. 
  8. [8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  9. [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  10. [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
  11. [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. 
  12. [12] G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1980. Zbl0020.29201
  13. [13] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990. 
  14. [14] Peter Jaeger. Elementary introduction to stochastic finance in discrete time. Formalized Mathematics, 20(1):1-5, 2012. doi:10.2478/v10037-012-0001-5. Zbl1276.91103
  15. [15] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990. 
  16. [16] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573-577, 1997. 
  17. [17] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990. 
  18. [18] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990. 
  19. [19] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990. 
  20. [20] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990. 
  21. [21] Bo Li, Yan Zhang, and Artur Korniłowicz. Simple continued fractions and their convergents. Formalized Mathematics, 14(3):71-78, 2006. doi:10.2478/v10037-006-0009-9. 
  22. [22] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997. 
  23. [23] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. 
  24. [24] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997. 
  25. [25] Christoph Schwarzweller. Proth numbers. Formalized Mathematics, 22(2):111-118, 2014. doi:10.2478/forma-2014-0013. 
  26. [26] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003. 
  27. [27] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990. 
  28. [28] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. 
  29. [29] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  30. [30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990. 
  31. [31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 
  32. [32] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Set sequences and monotone class. Formalized Mathematics, 13(4):435-441, 2005. 

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