New York Journal of Mathematics
Volume 3A (1997-1998) 117-124


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

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.

Author information

Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K.