The equality cases $P_t(\mathbb{N})=\frac{1}{2}$ for the deconvolved sum-of-digits measures

Abstract

Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in \mathbb{N}$, and let ${\mu }_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in \mathbb{Z}$. Let $P_t$ be the family of finitely supported measures defined by the convolution ${\mu }_t={\mu }_1\ast P_t$. Recently, Tarlowski (2026) has shown that the family $P_t$ may be represented as a recursively grown binary tree $T_t$, and that the Cusick's conjecture - ${\mu }_t(\mathbb{N})>\frac{1}{2}$, $t\in \mathbb{N}$, - follows from the asymmetry property of the family $T_t$, which was posed there as an open problem. Next, Cheng (2026) has provided the combinatorial description of the family $T_t$ in the language of principal subsequence ideals, and proved both conjectures. Both of these problems are directly related to the problem of determining the zeros of the function $\mathbb{N}\ni t\mapsto P_t(\mathbb{N})-\frac{1}{2}\in [0,\frac{1}{2}]$, a problem left open by Cheng (2026) as a saturation problem, and previously analyzed only numerically. In this paper we solve this problem completely. Writing an odd integer $t\ge 3$ as $t=(1w1{)}_2$ with $w\in \{0,1{\}}^{\star }$, we show that $P_t(\mathbb{N})=\frac{1}{2}$ if and only if $w$ is \emph{saturated} in the following sense: in the block decomposition $w=1^{a_0}01^{a_1}0\cdots 01^{a_k}$ with exactly $k$ zeros, every block of "1" satisfies $a_i\ge k$. Additionally, we show that the lower bound for $P_t(\mathbb{N})$ established by Cheng for $0$-initial words holds true for all non-saturated words.