On the sum-of-digits measures and Cusick's conjecture via stopped random walks

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 $\mu_t(d)$ denote the asymptotic density of the set of those natural numbers $n$ for which $s(n+t)-s(n)=d$. The $\mu_t$ are properly defined probability measures on $\Z$, and the Cusick conjecture states that $\mu_t(\mathbb{N})>\frac{1}{2}$ for any $t\in\mathbb{N}$. We investigate the properties of the family $\{\mu_t\}_{t\in\N}$ by reindexing the odd integers via a suitable partial order. This construction leads to a nonautonomous dynamics on pairs of probability measures on $\Z$, which represents the process of growing a tree. The associated stopped random walk allows a transparent structural description of those measures, including their support, symmetries, variance, and an asymptotic dichotomy between the central limit theorem and the almost sure convergence. Next, we focus on the median-preserving property of this process, and show that the Cusick conjecture is a special case of a more general claim about the asymmetric evolution of the associated binary trees, which we support numerically.