 

R. Nair
On Metric Diophantine Approximation and Subsequence Ergodic Theory


Published: 
March 27, 1998 
Keywords: 
metric diophantine approximation, continued fractions, subsequence ergodic theorems 
Subject: 
11K50,28D99 


Abstract
Suppose k_{n} denotes either
φ(n) or φ(p_{n}) (n = 1,2,... ) where the
polynomial φ maps the natural numbers to themselves
and p_{k} denotes the k^{th} rational prime. Let
(r_{n}/q_{n})_{n=1}^{∞} denote the sequence of
convergents to a real number x and define the the
sequence of approximation constants
(θ_{n}(x))_{n=1}^{∞} by
θ_{n}(x) = q_{n}^{2}∣ x  (r_{n}/q_{n})∣
(n = 1,2, ... ).
In this paper we study the behaviour of the sequence
(θ_{kn}(x))_{n=1}^{∞} for almost all x
with respect to Lebesgue measure. In the special case
where k_{n} = n (n = 1,2,... ) these
results are due to W. Bosma, H. Jager and F. Wiedijk.


Author information
Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K.
nair@liv.ac.uk

