The martingale evolution of probability measures defined via the sum-of-digits functions

Abstract: Let $s(n)$ denote the number of ones in the binary expansion of a natural number $n\in\mathbb{N}$. For any $t\in\mathbb{N}$ and $d\in\mathbb{Z}$, let $μt(d)$ denote the asymptotic density of the set of those natural numbers $n$ for which $s(n+t)-s(n)=d$. It is well known that $μ_t$ are properly defined probability measures on $\mathbb{Z}$, and the Cusick conjecture states that $μ_t(\mathbb{N})>\frac{1}{2}$ for any $t\in\mathbb{N}$. In this paper, we investigate the properties of the family ${μ_t}{t\in\mathbb{N}}$ by reindexing the odd integers via a suitable partial order. This construction leads to the nonautonomous dynamics on pairs of probability measures on $\mathbb{Z}$, and admits a natural interpretation in terms of evolution of planar binary trees and the corresponding stopping times. The measures $μ_t$ correspond to the marginal distributions of the associated stopped random walk. We will assume that the random walk starts from zero, and thus we will work with the family of measures $P_t$ determined by the convolution $μ_t=μ_1\ast P_t$. The martingale associated with the stopped random walk allows a transparent structural description of those measures, including their support, symmetries, variance, and the asymptotic behaviour. At the end we discuss the median preserving property of this martingale, and show that the Cusick conjecture is a special case of a more general claim about the asymmetric evolution of the binary trees associated to the martingale. This last claim is supported numerically at the end of the paper.