AI 中文总结
本文完全解决了Cheng提出的关于去卷积数位和测度族零点的饱和问题，给出奇数 $t$ 满足 $P_t(\mathbb{N})=\frac12$ 的充要条件，推广了Cheng的下界结论。
AI 中文摘要
设 $s(n)$ 表示整数 $n\in\mathbb{N}$ 的二进制展开中1的个数，$\mu_t$ 是定义在整数集 $\mathbb{Z}$ 上的概率测度，由函数 $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$ 的水平集的渐近密度给出。令 $P_t$ 为有限支撑测度族，满足卷积关系 $\mu_t=\mu_1*P_t$。近期Tarlowski（2026）证明了 $P_t$ 可表示为递归生长的二叉树 $T_t$，且Cusick猜想（对所有自然数 $t$，有 $\mu_t(\mathbb{N})>\frac12$）可由 $T_t$ 的非对称性导出，该非对称性被作为开放问题提出。随后Cheng（2026）用主子序列理想的语言给出了 $T_t$ 的组合描述并证明了上述两个猜想。这些问题均与确定函数 $\mathbb{N}\ni t\mapsto P_t(\mathbb{N})-\frac12\in[0,\tfrac12]$ 的零点直接相关，该问题被Cheng（2026）列为饱和问题，此前仅被数值分析过。本文完全解决了该问题：将奇数 $t\ge3$ 表示为 $t=(1\\,w\\,1)_2$（其中 $w\in\{0,1\}^\star$），证明 $P_t(\mathbb{N})=\frac12$ 当且仅当 $w$ 是“饱和”的，即对 $w=1^{a_0}\\,0\\,1^{a_1}\\,0\cdots0\\,1^{a_k}$（含恰好 $k$ 个0的块分解），每个“1”块满足 $a_i\ge k$；此外还证明Cheng针对以0开头的字建立的 $P_t(\mathbb{N})$ 下界对所有非饱和字均成立。
英文摘要
Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $μ_t$ 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 $μ_t=μ_1*P_t$. 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 - $μ_t(\mathbb{N})>\frac12$, $t\in\mathbb{N}$, - 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 $\mathbb{N}\ni t \mapsto P_t(\mathbb{N})-\frac12\in[0,\tfrac12]$, 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 $t\ge3$ as $t=(1\,w\,1)_2$ with $w\in\{0,1\}^{\star}$, we show that $P_t(\mathbb{N})=\frac12$ if and only if $w$ is \emph{saturated} in the following sense: in the block decomposition $w=1^{a_0}\,0\,1^{a_1}\,0\cdots0\,1^{a_k}$ 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.
