具有稠密轨道的积性函数的两点关联
Two-point correlations of multiplicative functions with dense orbits
AI总结：
本文研究具有稠密轨道的积性函数的两点关联，证明了满足特定条件的整数集合具有正下对数密度，强化了已有定理并提供了更简洁的证明。
AI中文摘要：
设$f,g:\mathbb{N}\to\mathbb{T}$是在复单位圆$\mathbb{T}$中具有稠密像的完全积性函数。我们证明，对于$\mathbb{T}^2$中每个非空开集$U$，使得$(f(n),g(n+1))\in U$的整数$n$的集合具有正的下对数密度，除非对$(f,g)$是特殊形式。该结果强化了Klurman、Mangerel以及Chamaras、Mountakis和Tsinas的早期定理，并给出了本质上更简单的证明。
英文摘要：
Let $f,g:\mathbb{N}\to\mathbb{T}$ be completely multiplicative functions with dense images in the complex unit circle $\mathbb{T}$. We prove that, for every non-empty open set $U \subset \mathbb{T}^2$, the set of integers $n$ such that $(f(n),g(n+1)) \in U$ has positive lower logarithmic density, unless the pair $(f,g)$ is of a special form. This result strengthens earlier theorems of Klurman and Mangerel, as well as of Charamaras, Mountakis, and Tsinas, and yields substantially simpler proofs.
