Lüroth Expansion Digits and Maclaurin’s Inequality

Abstract

It is well known that for almost all real number x, the geometric mean of the first n digits di (x) in the Luroth expansion of x converges to a number 0 K as n → ∞. On the other hand, for almost all x, the arithmetric mean of the first n Luroth expansion digits di (x) approaches infinity as n → ∞. 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≤k≤n. In this paper, we investigate what happens to the means of Luroth expansion digits 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 away from the arithmetic mean and convergence when one moves f(n) steps away from geometric mean.