Continued Fractions with Partial Quotients Bounded in Average

Abstract

We ask, for which $n$ does there exists a $k$, $1\le k<n$ and $(k,n)=1$, so that $k/n$ has a continued fraction whose partial quotients are bounded in average by a constant $B$? This question is intimately connected with several other well-known problems, and we provide a lower bound in the case of B=2.

7 pages, 0 figures; minor changes, to appear in Fibonacci Quarterly