Periodicity of Klein Polyhedra. Generalization of Lagrange’s Theorem
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The sails of algebraic multidimensional continued fractions possess combinatorial periodicity due to the action of the positive Dirichlet group on the sails. In case of one-dimensional geometric continued fractions this periodicity is completely described by the periodicity of the corresponding LLS sequences. The questions related to periodicity of multidimensional algebraic sails are important in algebraic number theory, since they are in correspondence with algebraic irrationalities. In particular, periods of algebraic sails characterize the groups of units in the corresponding orders.
This chapter is dedicated to such problems. First, we associate to any matrix with real distinct eigenvalues a multidimensional continued fraction. Second, we discuss periodicity of associated sails in the algebraic case. Further, we give examples of periods of two-dimensional algebraic continued fractions and formulate several questions arising in that context. Then we state and prove a version of Lagrange’s theorem for multidimensional continued fractions. Finally, we say a few words about the relation of the Littlewood and Oppenheim conjectures to periodic sails.
KeywordsContinue Fraction Infinite Series Golden Ratio Polygonal Line Algebraic Number Theory
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