Level sets of asymptotic mean of digits function for 4 -adic representation of real number

Abstract

We study topological, metric and fractal properties of the level sets Sθ = {x : r(x) = θ} of the function r of asymptotic mean of digits of a number x ∈ [0; 1] in its 4-adic representation, r(x) = lim n→∞ 1 n Xn i=1 αi(x) if the asymptotic frequency νj (x) of at least one digit does not exist, were νj (x) = lim n→∞ n −1#{k : αk(x) = j, k 6 n}, j = 0, 1, 2, 3.