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 M0. 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 j1. 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+ddet(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 j1. 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+ddet(M)L (θ).

@article{JTNB_1997__9_2_267_0,
     author = {Lagarias, Jeffrey C. and Shallit, Jeffrey O.},
     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},
     doi = {10.5802/jtnb.202},
     zbl = {0901.11024},
     mrnumber = {1617398},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/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. doi : 10.5802/jtnb.202. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_267_0/

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