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• W2951799891 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 $K = 2.6854520ldots$ (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 $1 leq k leq n$. 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." @default.
• W2951799891 created "2019-06-27" @default.
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• W2951799891 date "2014-02-02" @default.
• W2951799891 modified "2023-09-27" @default.
• W2951799891 title "Continued fraction digit averages an Maclaurin's inequalities" @default.
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• W2951799891 hasPublicationYear "2014" @default.
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