Continued fractions digit averages and MacLaurin inequalities

I would like to report on some very recent results on symmetric averages for continued fraction digits. Such averages are defined as the k-th root of the k-th elementary symmetric mean of the first n continued fraction digits of a real number. They interpolate between the classical arithmetic (k=1) and geometric (k=n) averages. We obtain sufficient conditions to ensure convergence / divergence of such means for typical real numbers and we provide nontrivial bound for large sets of of reals. Joint work with Doug Hensley, Steve J. Miller and Jake Wellens.