Continued fraction digit averages an Maclaurin's inequalities

Abstract

A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha )$ in the continued fraction expansion of $\alpha$ converges to a number $K=2.6854520\dots$ (Khinchin's constant) as $n\to \infty$. On the other hand, for almost all $\alpha$, the arithmetic mean of the first $n$ continued fraction digits $a_i(\alpha )$ approaches infinity as $n\to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1\le k\le n$. On the left end (when $k=n$) we have the geometric mean, and on the right end ($k=1$) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves $f(n)$ steps away from either extreme. We prove sufficient conditions on $f(n)$ to ensure to ensure divergence when one moves $f(n)$ steps away from the arithmetic mean and convergence when one moves $f(n)$ steps away from the geometric mean. For typical $\alpha$ we conjecture the behavior for $f(n)=cn$, $0<c<1$. We also study the limiting behavior of such means for quadratic irrational $\alpha$, providing rigorous results, as well as numerically supported conjectures.

arXiv comment32 pages, 7 figures. Substantial additions were made to previous version, including Theorem 1.3, Section 6, and Appendix A