Reversals and palindromes in continued fractions

Several results on continued fractions expansions are on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation and for formal power series.

[1]  Michel Dekking,et al.  Folds: III. More Morphisms , 1982 .

[2]  Pietro Corvaja,et al.  Some New Applications of the Subspace Theorem , 2002, Compositio Mathematica.

[3]  Aldo de Luca,et al.  Special factors, periodicity, and an application to Sturmian words , 2000, Acta Informatica.

[4]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[5]  Marina Ratner,et al.  Interactions Between Ergodic Theory, Lie Groups, and Number Theory , 1995 .

[6]  Aldo de Luca,et al.  Sturmian Words: Structure, Combinatorics, and Their Arithmetics , 1997, Theor. Comput. Sci..

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

[8]  J. Liouville.  Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques. , 1851 .

[9]  Jeffrey D. Vaaler,et al.  Review: Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra , 1991 .

[10]  Damien Roy.  Approximation simultanée d'un nombre et de son carré Simultaneous approximation to a real number and its square , 2003 .

[11]  A. Baker.  On Mahler's classification of transcendental numbers , 1964 .

[12]  Doron Zeilberger,et al.  Pythagorean primes and palindromic continued fractions. , 2005 .

[13]  Jean-Paul Allouche,et al.  On the conjectures of Rauzy and Shallit for infinite words , 1995 .

[14]  J. S. Frame.  Continued Fractions and Matrices , 1949 .

[15]  Jun-ichi Tamura.  Symmetric continued fractions related to certain series , 1991 .

[16]  M. Queffélec,et al.  Transcendance des fractions continues de Thue–Morse☆ , 1998 .

[17]  Aldo de Luca,et al.  Harmonic and gold Sturmian words , 2004, Eur. J. Comb..

[18]  Alan Baker,et al.  DIOPHANTINE APPROXIMATION (Lecture Notes in Mathematics, 785) , 1981 .

[19]  A. Baker,et al.  Continued fractions of transcendental numbers , 1962 .

[20]  Stéphane Fischler.  Palindromic prefixes and episturmian words , 2006, J. Comb. Theory, Ser. A.

[21]  S. Lang.  Number Theory III , 1991 .

[22]  Alfred J. van der Poorten.  Quadratic irrational integers with partly prescribed continued fraction expansion , 2004 .

[23]  Pierre Stambul,et al.  Continued fractions with bounded partial quotients , 1999 .

[24]  Alfred J. van der Poorten.  Specialisation and Reduction of Continued Fractions of Formal Power Series , 2004 .

[25]  Christoph Baxa,et al.  Extremal values of continuants and transcendence of certain continued fractions , 2004, Adv. Appl. Math..

[26]  Pierre Liardet,et al.  Séries de engel et fractions continuées , 2000 .

[27]  G. Paun,et al.  Jewels are Forever , 1999, Springer Berlin Heidelberg.

[28]  Wolfgang M. Schmidt,et al.  On simultaneous approximations of two algebraic numbers by rationals , 1967 .

[29]  G. Rauzy.  Suites à termes dans un alphabet fini , 1983 .

[30]  Henr. Jo. Smith.  De compositione numerorum primorum formae 4λ+1 ex duobus qudratis. , 1855 .

[31]  J. Davison,et al.  A series and its associated continued fraction , 1977 .

[32]  Grzegorz Rozenberg,et al.  Developments in Language Theory II , 2002 .

[33]  Jeffrey Shallit,et al.  Automatic Sequences: Theory, Applications, Generalizations , 2003 .

[34]  Jeffrey Shallit,et al.  ALGEBRAIC COMBINATORICS ON WORDS (Encyclopedia of Mathematics and its Applications 90) By M. LOTHAIRE: 504 pp., £60.00, ISBN 0 521 81220 8 (Cambridge University Press, 2002) , 2003 .

[35]  Walter Leighton,et al.  A general continued fraction expansion , 1939 .

[36]  Yann Bugeaud,et al.  Exponents of Diophantine Approximation and Sturmian Continued Fractions , 2004, math/0406064.

[37]  Wolfgang M. Schmidt,et al.  Approximation to real numbers by algebraic integers , 1969 .

[38]  Martine Queffélec,et al.  Approximations diophantiennes des nombres sturmiens , 2002 .

[39]  Alfred J. Van der Poorten,et al.  Symmetry and folding of continued fractions , 2002 .

[40]  Y. Bugeaud,et al.  On the complexity of algebraic numbers , II . Continued fractions by , 2006 .

[41]  Filippo Mignosi,et al.  Repetitions in the Fibonacci infinite word , 1992, RAIRO Theor. Informatics Appl..

[42]  Jeffrey Shallit,et al.  Simple continued fractions for some irrational numbers, II , 1979 .

[43]  William W. Adams,et al.  A remarkable class of continued fractions , 1977 .

[44]  J. Davison,et al.  Quasi-periodic continued fractions , 2007 .

[45]  A. Y. Khinchin.  Continued Fractions (In Russian) , 1937 .

[46]  Drew Vandeth,et al.  Sturmian words and words with a critical exponent , 2000, Theor. Comput. Sci..

[47]  Yann Bugeaud,et al.  On the Maillet-Baker continued fractions , 2005, math/0511681.

[48]  Wolfgang M. Schmidt,et al.  A theorem on linear forms , 1968 .

[49]  B. D. Mathan,et al.  Conjecture de Littlewood et récurrences linéaires , 2003 .

[50]  A. Katok,et al.  Invariant measures and the set of exceptions to Littlewood's conjecture , 2006, math/0612721.

[51]  Henrik L. Pedersen,et al.  Advanced Studies in Pure Mathematics , 2008 .

[52]  J. Littlewood.  Some problems in real and complex analysis , 1968 .

[53]  Yann Bugeaud,et al.  Palindromic continued fractions , 2005, math/0512014.

[54]  Zhi-Ying Wen,et al.  Some properties of the factors of Sturmian sequences , 2003, Theor. Comput. Sci..

[55]  Y. Bugeaud,et al.  On the complexity of algebraic numbers I. Expansions in integer bases , 2005, math/0511674.

[56]  Julien Cassaigne,et al.  Sequences with grouped factors , 1997, Developments in Language Theory.

[57]  R. C. Mason.  Norm form equations I , 1986 .

[58]  Andrew D. Pollington,et al.  On a problem in simultaneous diophantine approximation: Littlewood's conjecture , 2000 .

[59]  Jun-ichi Tamura,et al.  A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm , 1992 .

[60]  Damien Roy.  Approximation to real numbers by cubic algebraic integers I , 2002 .

[61]  Valérie Berthé,et al.  Initial powers of Sturmian sequences , 2006 .

[62]  H. Tietze.  Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions , 1908 .

[63]  Yann Bugeaud,et al.  A Short Proof of the Transcendence of Thue-Morse Continued Fractions , 2007, Am. Math. Mon..

[64]  Julien Cassaigne.  On a Conjecture of J. Shallit , 1997, ICALP.

[65]  J. Cassels,et al.  On the product of three homogeneous linear forms and indefinite ternary quadratic forms , 1951, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[66]  Yann Bugeaud,et al.  On the Littlewood conjecture in fields of power series , 2005 .

[67]  Françoise Dejean,et al.  Sur un Théorème de Thue , 1972, J. Comb. Theory A.

[68]  Y. Bugeaud,et al.  On the Littlewood Conjecture in Simultaneous Diophantine Approximation , 2005, math/0511678.

[69]  Edward B. Burger.  A Tail of Two Palindromes , 2005, Am. Math. Mon..

[70]  Filippo Mignosi,et al.  Infinite Words with Linear Subword Complexity , 1989, Theor. Comput. Sci..

[71]  Robert Kaufman,et al.  Continued fractions and Fourier transforms , 1980 .

[72]  J. Karhumäki,et al.  ALGEBRAIC COMBINATORICS ON WORDS (Encyclopedia of Mathematics and its Applications 90) By M. LOTHAIRE: 504 pp., 60.00, ISBN 0 521 81220 8 (Cambridge University Press, 2002) , 2003 .

[73]  P. Shiue,et al.  Substitution invariant cutting sequences , 1993 .

[74]  T. Cusick,et al.  The Markoff and Lagrange Spectra , 1989 .

[75]  David Damanik,et al.  Powers in Sturmian sequences , 2003, Eur. J. Comb..

[76]  Jeffrey Shallit,et al.  Folded continued fractions , 1992 .

[77]  M. Lothaire.  Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications) , 2005 .

[78]  E. Wirsing,et al.  Approximation mit algebraischen Zahlen beschränkten Grades. , 1961 .

[79]  Martine Queffélec,et al.  Irrational Numbers with Automaton-Generated Continued Fraction Expansion , 2000 .

[80]  K. F. Roth,et al.  Rational approximations to algebraic numbers , 1955 .

[81]  Aldo de Luca,et al.  Sturmian Words, Lyndon Words and Trees , 1997, Theor. Comput. Sci..

[82]  Damien Roy.  Approximation to real numbers by cubic algebraic integers. II , 2003 .

[83]  Julien Cassaigne,et al.  Limit Values of the Recurrence Quotient of Sturmian Sequences , 1999, Theoretical Computer Science.

[84]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[85]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[86]  Jean-Paul Allouche,et al.  Transcendence of Sturmian or Morphic Continued Fractions , 2001 .

[87]  V. Arnold,et al.  Mathematics: Frontiers and Perspectives , 2000 .

[88]  A. Khintchine.  Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen , 1924 .

[89]  Alfred J. van der Poorten,et al.  Formal Power Series and Their Continued Fraction Expansion , 1998, ANTS.

[90]  A. van der Poorten.  Beer and continued fractions with periodic periods , 1999 .

[91]  Jun-Ichi Tamura,et al.  Transcendental numbers having explicit $g$-adic and Jacobi-Perron expansions , 1992 .

[92]  D. Ridout.  Rational approximations to algebraic numbers , 1957 .

[93]  Jean Berstel,et al.  On the Index of Sturmian Words , 1999, Jewels are Forever.