Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Abstract

Recall Lebesgue’s singular function. Imagine flipping a biased coin with probability p of heads and probability q = 1 − p of tails. Let the binary expansion of ξ ∈ [0, 1]: ξ = ∑∞ k=1 ck2−k be determined by flipping the coin infinitely many times, that is, ck = 1 if the k-th toss is heads and ck = 0 if it is tails. We define Lebesgue’s singular function L(t) as the distribution function of the random variable ξ: L(t) = Prob{ξ < t}. It is well-known that L(t) is strictly increasing and its derivative is zero almost everywhere (p ̸= q). The moments of Lebesque’ singular function are defined as Mn = Eξn. The main result of this paper is the following: Mn = O(nlog2 p).