Institute of Geometry

This project studies topics which belong to both number theory and discrete geometry: more specifically, we investigated generalizations of concepts from Diophantine approximation. We followed a recently established line of research which links the approximation of maximal commutative subgroups on the one hand, and multidimensional continued fractions on the other hand. The aim of this project was to create efficient methodology concerning continued fractions in order to develop a kind of "approximation theory" for geometric primitives (lines, planes, simplicial cones, polyhedra) by rational objects defined in terms of appropriate integer lattices). This approach combines different notions of multidimensional continued fractions and their interpretation as maximal commutative subgroups.

The results obtained in the project include the following: Firstly we extended geometric Gauss reduction theory to the multidimensional case, in particular obtaining a complete set of geometric invariants (in terms of Klein-Voronoi continued fractions) of conjugacy classes of integer matrices. Another result is fast algorithms for the computation of finite Minkowski-Voronoi tessellations of the planes that correspond to rational maximal commutative subgroups (in the 3D case). Finally we we developed the approximation theory of arrangements of two 1-dimensional subspaces, showing an analogue of Lagrange's theorem on approximation rates.

This research has connections to other branches of mathematics. One link is to so-called limit shape problems of Young diagrams or convex lattice polygons, when they are considered in the simplicial cone setting. Another link is to the approximation by rational cones which correspond to singularities of complex toric varieties.

See [here] for an earlier project page on multidimensional continued fractions. Work on this project in the years 2011–2013 was carried out at TU Graz.

• O. Karpenkov. Geometry of continued fractions. Springer, 2013. ISBN 978-3-642-39367-9. [doi].
• O. Karpenkov. Multidimensional Gauss reduction theory for conjugacy classes of SL(n,Z). J. Th. Nombres Bordeaux 25 (2013), 99-109. [MR], [doi].
• O. Karpenkov and A. Ustinov. Geometry of Minkowski-Voronoi tessellations of the plane. Technical Report 1407.0135, ArXiv, 2014-2017. [arxiv:1407.0135].
• O. Karpenkov. On asymptotic reducibility in SL(3,mathbb Z). Technical Report 1205.4166, ArXiv, 2012. [arxiv:1205.4166].
• O. N. Karpenkov and A. M. Vershik. Rational approximation of maximal commutative subgroups of GL(n,R). J. Fixed Point Theory Appl. 7/1 (2010), 241-263. [MR], [doi].
• O. Karpenkov. Continued fractions and the second Kepler law. Manuscripta Mathematica 134 (2011), 157-169. [MR], [doi], [arxiv:0911.2791].
• O. Karpenkov. On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators. Math. Notes 88/1-2 (2010), 28-38, Russian version: Mat. Zametki, 88(1), (2010), 30--42. [doi].
• O. Karpenkov. On irrational lattice angles. Funct. Anal. Other Math. 2/2-4 (2009), 221-239. [Zbl], [MR], [doi].
• O. Karpenkov. Constructing multidimensional periodic continued fractions in the sense of Klein. Math. Comp. 78/267 (2009), 1687-1711. [MR], [doi], [arxiv:math/0411031].
• O. Karpenkov. Elementary notions of lattice trigonometry. Math. Scand. 102/2 (2008), 161-205. [Zbl], [MR].
• O. Karpenkov. Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions. Monatsh. Math. 152/3 (2007), 217-249. [Zbl], [MR], [doi], [arxiv:math/0510482].
• O. Karpenkov. On a criterion for the existence and uniqueness of an integer triangle with given angles. Russian Math. Surveys 61/6 (2006), 1178-1179. [Zbl], [MR], [doi].
• O. Karpenkov. Three examples of three-dimensional continued fractions in the sense of Klein. C. R. Math. Acad. Sci. Paris 343/1 (2006), 5-7. [Zbl], [MR], [doi], [arxiv:math/0601493].
• O. Karpenkov. Classification of three-dimensional multistoried completely hollow convex marked pyramids. Russian Math. Surveys 60 (2005), 165-166. [Zbl], [MR], [doi].
• O. Karpenkov. On two-dimensional continued fractions of hyperbolic integer matrices with small norm. Russian Math. Surveys 59/5 (2004), 959-960. [Zbl], [MR], [doi].
• O. Karpenkov. On the triangulations of tori associated with two-dimensional continued fractions of cubic irrationalities. Funct. Anal. Appl. 38/2 (2004), 102-110, Russian version: Funkt. Anal. Prilozh. 38 (2004), 28-37. [Zbl], [MR], [doi].

• V. I. Arnold. Continued fractions. Moscow Center of Continuous Mathematical Education, Moscow, 2002.
• V. I. Arnold. Higher-dimensional continued fractions. Regul. Chaotic Dyn. 3/3 (1998), 10-17, J. Moser at 70 (Russian). [MR].
• O. N. German. Sails and Hilbert bases. Tr. Mat. Inst. Steklova 239/Diskret. Geom. i Geom. Chisel (2002), 98-105. [MR].
• F. Klein. Ueber eine geometrische Auffassung der gewöhnlichen Kettenbruchentwicklung. Nachr. Ges. Wiss. Göttingen Math-Phys. Kl. 3/3 (1895), 352-357.
• F. Klein. Sur une représentation géométrique de développement en fraction continue ordinaire. Nouv. Ann. Math. 15/3 (1896), 327-331.
• M. L. Kontsevich and Y. M. Suhov. Statistics of Klein polyhedra and multidimensional continued fractions. In Pseudoperiodic topology, volume 197 of Amer. Math. Soc. Transl. Ser. 2, pages 9-27. Amer. Math. Soc., Providence, RI, 1999. [MR].
• E. I. Korkina. The simplest 2-dimensional continued fraction. J. Math. Sci. 82/5 (1996), 3680-3685, Topology, 3. [MR].
• E. I. Korkina. Two-dimensional continued fractions. The simplest examples. Trudy Mat. Inst. Steklov. 209/Osob. Gladkikh Otobrazh. s Dop. Strukt. (1995), 143-166. [MR].
• E. Korkina. La périodicité des fractions continues multidimensionnelles. C. R. Acad. Sci. Paris Sér. I Math. 319/8 (1994), 777-780. [MR].
• G. Lachaud. Voiles et polyhedres de Klein. Act. Sci. Ind. Hermann, 2002. 176 pp.
• Z.-O. Mussafir. Sails and Hilbert bases. Funktsional. Anal. i Prilozhen. 34/2 (2000), 43-49, 96. [MR].
• H. Tsuchihashi. Higher-dimensional analogues of periodic continued fractions and cusp singularities. Tohoku Math. J. (2) 35/4 (1983), 607-639. [MR].