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

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).

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).

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     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},
     year = {1997},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     zbl = {0901.11024},
     mrnumber = {1617398},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_267_0/}
}
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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/

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