On continued fraction algorithms

Is there a good continued fraction approximation between every two bad ones? What is the entropy of the natural extension for alpha-Rosen fractions? How do you find multi-dimensional continued fractions with a guaranteed quality in polynomial time? These, and many more, questions are answered in this thesis.

[1]  H. Nakada.  Metrical Theory for a Class of Continued Fraction Transformations and Their Natural Extensions , 1981 .

[2]  R. Forcade,et al.  Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two , 1979 .

[3]  F. Schweiger.  Ergodic Theory of Fibred Systems and Metric Number Theory , 1995 .

[4]  Donald S. Ornstein,et al.  Ornstein theory , 2008, Scholarpedia.

[5]  C. Series,et al.  The Hurwitz Constant and Diophantine Approximation on Hecke Groups , 1986 .

[6]  Fritz Schweiger,et al.  Multidimensional continued fractions , 2000 .

[7]  J. Tong.  Approximation by Nearest Integer Continued Fractions. , 1992 .

[8]  H. Alzer.  On segre’s theorem on asymmetric diophantine approximation , 1997 .

[9]  W. Bosma,et al.  Metrical theory for optimal continued fractions , 1990 .

[10]  Mukarram Ahmad,et al.  Continued fractions , 2019, Quadratic Number Theory.

[11]  H. Nakada.  On the Lenstra constant associated to the Rosen continued fractions , 2007, 0705.3756.

[12]  Jeffrey C. Lagarias,et al.  The computational complexity of simultaneous Diophantine approximation problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[13]  M. Nathanson.  Approximation by continued fractions , 1974 .

[14]  J. Lagarias.  Geodesic Multidimensional Continued Fractions , 1994 .

[15]  E. H. Neville.  Mathematische Werke , 1948, Nature.

[16]  Jacobi,et al.  Allgemeine Theorie der kettenbruchaehnlichen Algorithmen , 1869 .

[17]  Bettina Just.  Generalizing the Continued Fraction Algorithm to Arbitrary Dimensions , 1992, SIAM J. Comput..

[18]  D. Rosen.  A class of continued fractions associated with certain properly discontinuous groups , 1954 .

[19]  Phong Q. Nguyen,et al.  The LLL Algorithm - Survey and Applications , 2009, Information Security and Cryptography.

[20]  F. Wiedijk,et al.  Some metrical observations on the approximation by continued fractions , 1983 .

[21]  Shunji Ito,et al.  ON THE INVARIANT MEASURE FOR THE TRANSFORMATIONS ASSOCIATED WITH SOME REAL CONTINUED-FRACTIONS. , 1977 .

[22]  H. Nakada,et al.  The non-monotonicity of the entropy of α-continued fraction transformations , 2008 .

[23]  Laura Luzzi,et al.  On the entropy of Japanese continued fractions , 2006 .

[24]  A. Hurwitz,et al.  Ueber die angenäherte Darstellung der Zahlen durch rationale Brüche , 1894 .

[25]  Jeffrey C. Lagarias,et al.  Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators , 1982 .

[26]  W. Doeblin.  Remarques sur la théorie métrique des fractions continues , 1940 .

[27]  J. Tong.  The conjugate property of the Borel theorem on diophantine approximation , 1983 .

[28]  C. Kraaikamp,et al.  On the approximation by continued fractions, II , 1989 .

[29]  Diophantine approximation by continued fractions , 1991 .

[30]  O. Perron,et al.  Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus , 1907 .

[31]  Arnaldo Nogueira,et al.  Multidimensional Continued Fractions. By Fritz Schweiger. Oxford Science Publications , 2002, Ergodic Theory and Dynamical Systems.

[32]  Thomas A. Schmidt,et al.  Natural extensions for the Rosen fractions , 1999 .

[33]  M. Iosifescu,et al.  Metrical Theory of Continued Fractions , 2002 .

[34]  Meir Smorodinsky,et al.  Bernoulli schemes of the same entropy are finitarily isomorphic , 1979 .

[35]  V. Rohlin,et al.  Exact endomorphisms of a Lebesgue space , 1961 .

[36]  F. Bagemihl,et al.  Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions. , 1966 .

[37]  Thomas A. Schmidt,et al.  Metric and arithmetic properties of mediant-Rosen maps , 2008, 0812.0548.

[38]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[39]  P. Shields.  The Ergodic Theory of Discrete Sample Paths , 1996 .