Linear fractional transformations of continued fractions with bounded partial quotients
Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 267-279.

Soit θ un nombre réel de développement en fraction continue

θ=a 0 ,a 1 ,a 2 ,⋯,
et soit
M=abcd
une matrice d’entiers tel que det M≠0. Si θ est à quotients partiels bornés, alors aθ+b cθ+d=a 0 * ,a 1 * ,a 2 * ,⋯ est aussi à quotients partiels bornés. Plus précisément, si a j ≤K pour tout j suffisamment grand, alors a j * ≤|det(M)|(K+2) pour tout j suffisamment grand. Nous donnons aussi une borne plus faible qui est valable pour tout a j * avec j≥1. Les démonstrations utilisent la constante d’approximation diophantienne homogène L ∞ θ=lim sup q→∞ qq θ -1 . Nous montrons que
1 det(M)L ∞ (θ)≤L ∞ aθ+b cθ+d≤det(M)L ∞ (θ).

Let θ be a real number with continued fraction expansion

θ=a 0 ,a 1 ,a 2 ,⋯,
and let
M=abcd
be a matrix with integer entries and nonzero determinant. If θ has bounded partial quotients, then aθ+b cθ+d=a 0 * ,a 1 * ,a 2 * ,⋯ also has bounded partial quotients. More precisely, if a j ≤K for all sufficiently large j, then a j * ≤|det(M)|(K+2) for all sufficiently large j. We also give a weaker bound valid for all a j * with j≥1. The proofs use the homogeneous Diophantine approximation constant L ∞ θ=lim sup q→∞ qq θ -1 . We show that
1 det(M)L ∞ (θ)≤L ∞ aθ+b cθ+d≤det(M)L ∞ (θ).

@article{JTNB_1997__9_2_267_0,
     author = {J. C. Lagarias and J. O. Shallit},
     title = {Linear fractional transformations of continued fractions with bounded partial quotients},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {267--279},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     zbl = {0901.11024},
     mrnumber = {1617398},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_267_0/}
}
TY  - JOUR
AU  - J. C. Lagarias
AU  - J. O. Shallit
TI  - Linear fractional transformations of continued fractions with bounded partial quotients
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1997
SP  - 267
EP  - 279
VL  - 9
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_267_0/
LA  - en
ID  - JTNB_1997__9_2_267_0
ER  - 
%0 Journal Article
%A J. C. Lagarias
%A J. O. Shallit
%T Linear fractional transformations of continued fractions with bounded partial quotients
%J Journal de théorie des nombres de Bordeaux
%D 1997
%P 267-279
%V 9
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_267_0/
%G en
%F JTNB_1997__9_2_267_0
J. C. Lagarias; J. O. Shallit. Linear fractional transformations of continued fractions with bounded partial quotients. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 267-279. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_267_0/

1 A. Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984. | MR | Zbl

2 A. Chitelet, Contribution à la théorie des fractions continues arithmétiques, Bull. Soc. Math. France 40 (1912), 1-25. | JFM | Numdam | MR

3 S.D. Chowla, Some problems of diophantine approximation (I), Math. Zeitschrift 33 (1931), 544-563. | JFM | MR | Zbl

4 T.W. Cusick and M. Flahive, The Markoff and Lagrange Spectra, American Mathematical Society, Providence, RI, 1989. | MR | Zbl

5 T.W. Cusick and M. Mendès France, The Lagrange spectrum of a set, Acta Arith. 34 (1979), 287-293. | MR | Zbl

6 H. Davenport, A remark on continued fractions, Michigan Math. J. 11 (1964), 343-344. | MR | Zbl

7 M. Hall, On the sum and product of continued fractions, Annals of Math. 48 (1947), 966-993. | MR | Zbl

8 G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press. | MR

9 A. Hurwitz, Über die angenäherte Darstellungen der Zahler durch rationale Brüche, Math. Ann. 44 (1894), 417-436. | JFM | MR

10 D.E. Knuth, The Art of Computer Programming, Vol. II: Seminumerical Algorithms, Addison-Wesley, 1981. | MR | Zbl

11 M. Mendès France, Sur les fractions continues limitées, Acta Arith. 23 (1973), 207-215. | MR | Zbl

12 M. Mendès France, The depth of a rational number, Topics in Number Theory (Proc. Colloq. Debrecen, 1974) Colloq. Soc. Janos Bolyai, vol. 13, North-Holland, Amsterdam, 1976, pp. 183-194. | MR | Zbl

13 M. Mendès France, On a theorem of Davenport concerning continued fractions, Mathematika 23 (1976), 136-141. | MR | Zbl

14 O. Perron, Über die Approximation irrationaler Zahlen durch rationale" Sitz. Heidelberg. Akad. Wiss. XIIA (4. Abhandlung) (1921), 3-17. | JFM

15 G.N. Raney, On continued fractions and finite automata, Math. Annalen 206 (1973), 265-283. | MR | Zbl

16 W. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, vol. 785, Springer-Verlag, 1980. | MR | Zbl

17 J.O. Shallit, Continued fractions with bounded partial quotients: a survey, Enseign. Math. 38 (1992), 151-187. | Zbl

18 H.M. Stark, Introduction to Number Theory, Markham, 1970. | MR | Zbl