Continued Fractions

Some (mostly on-line) resources

Books:

• O. Karpenkov, Geometry of continued Fractions, Springer,2013 ( library link).
• G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, Oxford : Clarendon Press, 1979( library link).
• A. Hatcher, Topology of numbers, unpublished draft.

Lecture notes:

• O. Karpenkov, Continued Fractions and Generalizations Lecture notes, TU Graz, 2011.
• W. F. Hammond, Continued Fractions and the Euclidean Algorithm. Lecture notes, University at Albany, 1997.
• G. G. Krishnan, Continued Fractions. Lecture notes, Cornell University, 2016.

Introductory papers:

• Caroline Series, The Geometry of Markoff Numbers.
• C. Series, Continued fractions and hyperbolic geometry.
• M. C. Irwin, Geometry of continued fractions.
• I. Richards, Continued frections without tears.
• J. Barrow, Chaos in Numberland: The secret life of continued fractions.

Presentations:

• A. Leonardis, Simple applications of Continued Fractions and an elementary result on Heron’s algorithm .
• M. Walker, Continued fractions and hyperbolic geometry.
• M. Waldschmid, Continued fractions: introduction and applications.
• A. Veselov, Arithmetic and geometry of Conway river.
• I. Short, Geodesic continued fractions. See also other presentations by Ian Short.

Webpages:

• R. Knott, Continued fractions. ----
• Y. Grabovsky, If Pope Gregory new continued fractions. ----

Some recent results:

• I. Short, M. Walker Even-integer continued fractions and the Farey tree.
• I. Canakci, R. Schiffler Snake graphs and continued fractions.
• L. H. Kauffman and S. Lambropoulou From tangle frections to DNA.
• K. Spalding, A. Veselov, Conway river and Arnold sail.
• V. Ovsienko, Partitions of unity in SL(2,Z), negative continued fractions, and dissections of polygons.
• O. Karpenkov, Continued Fraction approach to Gauss Reduction Theory.
• O. Karpenkov, On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups.

More papers:

• D. J. Grabiner, Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions.
• R. Moeckel, Geodesics on modular surfaces and continued fractions.

Some slides:

• Continued Fractions and SL2-tilings LMS Northern Regional Meeting and Workshop, slides