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On approximation by Lüroth series

Karma Dajani, Cor Kraaikamp (1996)

Journal de théorie des nombres de Bordeaux

Let x ] 0 , 1 ] and p n / q n , n 1 be its sequence of Lüroth Series convergents. Define the approximation coefficients θ n = θ n ( x ) by q n x - p n , n 1 . In [BBDK] the limiting distribution of the sequence ( θ n ) n 1 was obtained for a.e. x using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each n , θ n is already distributed according to the limiting distribution. Using the natural extension we will study the distribution for...

On inhomogeneous diophantine approximation with some quasi-periodic expressions, II

Takao Komatsu (1999)

Journal de théorie des nombres de Bordeaux

We consider the values concerning ( θ , φ ) = lim inf | q | | q | | | q θ - φ | | where the continued fraction expansion of θ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying ( θ , φ ) = 0 .

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