Geometry of Continued Fractions pp 115-136 | Cite as

# Geometric Aspects of Approximation

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## Abstract

The approximation properties of continued fractions have attracted researchers for centuries. There are many different directions of investigation in this important subject (the study of best approximations, badly approximable numbers, etc.). In this chapter we consider two geometric questions of approximations by continued fractions. First, we prove two classical results on best approximations of real numbers by rational numbers. Second, we describe a rather new branch of generalized Diophantine approximations concerning arrangements of two lines in the plane passing through the origin. In this chapter we use some material related to basic properties of continued fractions of Chap. 1, to geometry of numbers of Chap. 3, and to Markov–Davenport forms of Chap. 7.

## Keywords

Continued Fraction Geometric Questions Diophantine Approximation Integer Arrangements Eigenlines## References

- 101.O.N. Karpenkov, A.M. Vershik, Rational approximation of maximal commutative subgroups of \(\operatorname{GL}(n,\Bbb{R})\). J. Fixed Point Theory Appl.
**7**(1), 241–263 (2010) MathSciNetzbMATHCrossRefGoogle Scholar - 127.J.C. Lagarias, Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators. Trans. Am. Math. Soc.
**272**(2), 545–554 (1982) MathSciNetzbMATHGoogle Scholar