Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published January 30, 2015 | public
Journal Article Open

Continued Fraction Digit Averages and Maclaurin's Inequalities


A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits a_i(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.

Additional Information

© 2015 Taylor & Francis Group, LLC. Published online: 30 Jan 2015. We thank Iddo Ben-Ari and Keith Conrad for sharing the preprint [Ben-Ari and Conrad 14] with us, Xiang-ShengWang for mentioning formula (4–11) to us, and Harold G. Diamond for useful discussions. A special acknowledgment is due to Jairo Bochi, who pointed out to us the important reference [Halász and Székely 76]. Francesco Cellarosi was partially supported by an AMS-Simons travel grant and NSF grant DMS1363227, Steven Miller was supported by NSF grants DMS0970067 and DMS1265673, and Jake Wellens was supported by NSF grant DMS0850577 and Williams College.

Attached Files

Submitted - 1402.0208v3.pdf


Files (1.4 MB)
Name Size Download all
1.4 MB Preview Download

Additional details

August 20, 2023
August 20, 2023