Titlebook: Geometry of Continued Fractions; Oleg N. Karpenkov Textbook 2022Latest edition Springer-Verlag GmbH Germany, part of Springer Nature 2022

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Geometry of Regular Continued Fractionstinued fractions in terms of integer lengths of edges and indices of angles for the boundaries of convex hulls of all integer points inside certain angles. In the next chapter we will extend this construction to construct a complete invariant of integer angles. For the geometry of continued fraction

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Complete Invariant of Integer Anglesger angles, constructing a certain integer broken line called the . of an angle. We combine the integer invariants of a sail into a sequence of positive integers called an .. From one side, the notion of LLS sequence extends the notion of continued fraction (see Remark 4.8), about which we will say

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Integer Trigonometry for Integer Anglestrigonometry has many similarities to Euclidean trigonometry (for instance, integer arctangents coincide with real arctangents; the formulas for adjacent angles are similar). From another point of view they are totally different, since integer sines and cosines are positive integers; there are two r

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Minima of Quadratic Forms, the Markov Spectrum and the Markov-Davenport Characteristics (excluding the origin). In this chapter we briefly discuss this classical subject, focusing on the discrete Markov spectrum that has the most relevant connection to geometry of continued fractions. We conclude this chapter with the notion of Markov—Davenport characteristic that we use later in the

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Continuant Representation of GL(2, ℤ) Matrices of the modular group (see Section 9.1). It turns out that SL(2, .) can be generated by two elements. Further we discuss an alternative approach to represent elements of GL(2, .) (which includes SL(2, .)) that is based on continuant representation of matrices.

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Textbook 2022Latest editiondiverse areas of mathematics..The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses..

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Funktionen stochastischer Größentinued fractions in terms of integer lengths of edges and indices of angles for the boundaries of convex hulls of all integer points inside certain angles. In the next chapter we will extend this construction to construct a complete invariant of integer angles. For the geometry of continued fraction