Continued Fraction Digit Averages and Maclaurin’s Inequalities

Abstract

A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520… (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin’s inequalities, relating the 1/kth powers of the kth elementary symmetric means of n numbers for 1 ≤ k ≤ 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 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. We show for almost all α and appropriate k as a function of n that S(α, n, k)1/k is of order log (n/k). For typical α, we find the limit for f(n) = cn, 0 < c < 1. We also study the limiting behavior of such means for quadratic irrational α, providing rigorous results, as well as numerically supported conjectures.