Continued fraction digit averages and MacLaurin's Inequalities

Continued fraction digit averages and MacLaurin's Inequalities

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Francesco Cellarosi, University of Illinois, Urbana-Champaign
Fine Hall 322

Please note different day and location.  A classical result of Kinchin says that for almost every real number x, the geometric mean of the first n digits in the continued fraction expansion of x converges to a number K=2.685... as n tends to infinity. On the other hand, for almost every x, the arithmetic mean of the first n digits tends to infinity. There is a sequence of refinements of the classical Arithmetic Mean - Geometric Mean inequality (called MacLaurin's inequalities) involving the k-th root of the k-th elementary symmetric mean, where k ranges from 1 (arithmetic mean) to n (geometric mean). We analyze what happens to these means for typical real numbers, when k is a function of n. We obtain sufficient conditions to ensure convergence / divergence of such means.   Joint work with Steven J. Miller and Jake L. Wellens