Geometry of Continued Fractions by Oleg Karpenkov

By Oleg Karpenkov

Traditionally a topic of quantity thought, persevered fractions look in dynamical platforms, algebraic geometry, topology, or even celestial mechanics. the increase of computational geometry has ended in renewed curiosity in multidimensional generalizations of endured fractions. a number of classical theorems were prolonged to the multidimensional case, casting gentle on phenomena in diversified parts of arithmetic. This e-book introduces a brand new geometric imaginative and prescient of endured fractions. It covers numerous purposes to questions relating to such parts as Diophantine approximation, algebraic quantity conception, and toric geometry.

The reader will locate an outline of present development within the geometric idea of multidimensional persisted fractions followed by way of at present open difficulties. each time attainable, we illustrate geometric buildings with figures and examples. every one bankruptcy has routines necessary for undergraduate or graduate courses.

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The same is true in lattice geometry. 5 Two integer segments are congruent if and only if they have the same integer length. Proof Let AB be an integer segment of length k. Let us prove that it is integer congruent to the integer segment with endpoints O(0, 0) and K(k, 0). Consider an integer translation sending A to the origin O. Let this translation send B to an integer point B (x, y). Since l (OB ) = l (AB) = k, we have x = kx and y = ky , where x and y are relatively prime. Hence there exists an integer pair (a, b) such that bx − ay = 1.

As in Euclidean geometry, integer area does not uniquely determine the congruence class of triangles. Nevertheless, all integer triangles of unit integer area are congruent, since the vectors of the edges of such triangles generate the integer lattice. 7 Index of Rational Angles An angle is called rational if its vertex is an integer point and both its edges contain integer points other than the vertex. 8 The index of a rational angle ∠BAC, denoted by lα(∠AOB), is the index of the sublattice generated by all integer vectors of the lines AB and AC in the integer lattice.

6 we have that the elements of the LLS sequence for larctan s coincide with the elements of the regular continued fraction for s. Hence the statement holds by definition of the integer tangent. (ii) Both angles larctan(ltan α) and α have the same LLS sequences. 10. In the following proposition we collect several trigonometric properties. 5 (i) The values of integer trigonometric functions for integer congruent angles coincide. (ii) The lattice sine and cosine of any rational angle are relatively prime positive integers.

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