The Möbius transformation of continued fractions with bounded upper and lower partial quotients

Let h: x 7→ ax+b cx+d be the nondegenerate Möbius transformation with integer entries. We get a bound of the continued fraction of h x by upper and lower bounds of the continued fraction of x.

Keywords:

Möbius transformation, continued fraction expansion partial quotient.,

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[1] Avila A, Krikorian R. Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Annals of Mathematics. Second Series 2006; 164 (3): 911-940.
[2] Badziahin D, Bugeaud Y, Einsiedler M, Kleinbock D. On the complexity of a putative counterexample to the p-adic Littlewood conjecture. Compositio Mathematica 2015; 151 (9): 1647-1662.
[3] Cusick TW, Mendès France M. The Lagrange spectrum of a set. Acta Arithmetica 1979; 34 (4): 287-293.
[4] Einsiedler M, Fishman L, Shapira U. Diophantine approximations on fractals. Geometric and Functional Analysis 2011; 21 (1): 14-35.
[5] Hall M. On the sum and product of continued fractions. Annals of Mathematics. Second Series 1947; 48: 966-993.
[6] Hines R. Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation. PhD thesis, University of Colorado at Boulder, 2019.
[7] Lagarias JC, Shallit JO. Correction to: “Linear fractional transformations of continued fractions with bounded partial quotients” [J. Théor. Nombres Bordeaux 9 (1997), no. 2, 267-279; mr1617398]. Journal de Théorie des Nombres de Bordeaux 2003; 15 (3): 741-743.
[8] Last Y. Zero measure spectrum for the almost Mathieu operator. Communications in Mathematical Physics 1994; 164 (2): 421-432.
[9] Liardet P, Stambul P. Algebraic computations with continued fractions. Journal of Number Theory 1998; 73 (1): 92-121.
[10] Řada H, Starosta Š. Bounds on the period of the continued fraction after a möbius transformation. Journal of Number Theory (to appear).
[11] Raney GN. On continued fractions and finite automata. Mathematische Annalen 1973; 206: 265-283.
[12] Shallit J. Real numbers with bounded partial quotients: a survey. L’Enseignement Mathématique. Revue Internationale. 2e Série 1992; 38 (1-2): 151-187.
[13] Simon B. Schrödinger operators in the twenty-first century. Mathematical physics 2000 Imp. Coll. Press, London 2000, 283-288.
[14] Stambul P. Continued fractions with bounded partial quotients. Proceedings of the American Mathematical Society 2000; 128 (4): 981-985.