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R. Nair
On Metric Diophantine Approximation and Subsequence Ergodic Theory
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| Published: |
March 27, 1998 |
| Keywords: |
metric diophantine approximation, continued fractions, subsequence ergodic theorems |
| Subject: |
11K50,28D99 |
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Abstract
Suppose kn denotes either
φ(n) or φ(pn) (n = 1,2,... ) where the
polynomial φ maps the natural numbers to themselves
and pk denotes the kth rational prime. Let
(rn/qn)n=1∞ denote the sequence of
convergents to a real number x and define the the
sequence of approximation constants
(θn(x))n=1∞ by
θn(x) = qn2∣ x - (rn/qn)∣
(n = 1,2, ... ).
In this paper we study the behaviour of the sequence
(θkn(x))n=1∞ for almost all x
with respect to Lebesgue measure. In the special case
where kn = n (n = 1,2,... ) these
results are due to W. Bosma, H. Jager and F. Wiedijk.
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| Author information
Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K.
nair@liv.ac.uk
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