AI 中文总结
研究自然数集子集沿规定福勒序列的加法与乘法密度，证明加法上密度为1时沿合适乘法福勒序列乘法密度为1及独立性，通过素数赋值坐标得到密度公式和随机模型，还将乘法相关性应用于符号动力系统并给出遍历性和混合准则。
AI 中文摘要
我们研究了自然数集子集沿规定的福勒序列的加法和乘法密度。证明了加法上密度为1意味着沿合适的乘法福勒序列乘法密度为1。还证明了独立性：给定加法福勒序列$(G_n)_n$、乘法福勒序列$(F_n)_n$以及任意$(\alpha,\beta)\in[0,1]^2$，构造单个集合$A\subseteq\N$使得$\dens_{(G_n)_n}(A)=\alpha$且$\md_{(F_n)_n}(A)=\beta$。素数赋值坐标给出了一个局部条件的精确密度公式和随机模型，在可和性假设下给出可数多个；在有限坐标情况下还给出精确的高阶相关性。最后，将这些乘法相关性实现为符号动力系统中的相关性并获得遍历性和混合的准则。
英文摘要
We study additive and multiplicative densities of subsets of $\N$ along prescribed Følner sequences. We prove that additive upper density one implies multiplicative density one along a suitable multiplicative Følner sequence. We also prove independence in the following sense: given an additive Følner sequence $(G_n)_n$, a multiplicative Følner sequence $(F_n)_n$, and any $(α,β)\in[0,1]^2$, we construct a single set $A\subseteq\N$ such that $\dens_{(G_n)_n}(A)=α$ and $\md_{(F_n)_n}(A)=β$. Prime-valuation coordinates yield exact density formulas and random models for one local condition and, under summability assumptions, countably many; in the finite-coordinate case they also give exact higher-order correlations. Finally, we realize these multiplicative correlations as correlations in symbolic dynamical systems and obtain criteria for ergodicity and mixing.
