New York Journal of Mathematics
Volume 11 (2005) 345-349


Andrew Haas

An ergodic sum related to the approximation by continued fractions

Published: July 21, 2005
Keywords: Continued fractions, metric theory, interval maps
Subject: 11J70, 11J83, 37E05

To each irrational number x is associated an infinite sequence of rational fractions (pn/qn), known as the convergents of x. Consider the functions
qn |qnx-pn |=θn(x).
We shall primarily be concerned with the computation, for almost all real x, of the ergodic sum
limn→∞ (1/n)∑k=1nlogθk(x)= -1-(1/2)log 2≈ -1.34657.

Author information

Department of Mathematics, The University of Connecticut, Storrs, CT. 06269-3009