AI 中文总结
研究巴拿赫空间子集模贫集的线性关系，通过证明拓扑零一律，给出子集\(A\)与\(A + x\)在模贫集意义下相等的条件，以及相关等式成立的条件，得到了一些结果的范畴类似物。
AI 中文摘要
我们证明了几个拓扑零一律。首先，若巴拿赫空间\(X\)的子集\(A\)具有贝尔性质，则\(A\)存在某个向量的稠密集\(x\in X\)，使得\(A + x\)与\(A\)在模贫集意义下相等当且仅当\(A\)要么是贫集要么是余贫集。当\(X = \mathbb{R}\)时给出了其他等价条件。其次，证明若\(A_1,\ldots,A_k\subseteq \mathbb{R}\)是具有贝尔性质的子集，\(\alpha_1,\ldots,\alpha_k\)是非零实数且有限和不同，\((x_{n,i}: n\in \omega)\)是对每个\(i = 1,\ldots,k\)都收敛到\(0\) 的单射实数列，那么\(\sum_{i=1}^k \alpha_i(1_{A_i+x_{n,i}}-1_{A_i})=0\)在模贫集意义下对所有\(n \in \omega\)成立当且仅当每个\(A_i\)要么是贫集要么是余贫集。最后，在\(\alpha_1,\ldots,\alpha_k\)没有不同有限和的情况下给出了一些额外结果。这些结果产生了Fejzić、Freiling和Rinne在[J. London Math. Soc. 82 (2010), no. 3, 717--732]中几个结果的范畴类似物。
英文摘要
We prove several topological zero-one laws. First, we show that, if a subset $A$ of a Banach space $X$ has the Baire property, then $A$ admits a somewhere dense set of vectors $x \in X$ such that $A+x$ agrees with $A$ modulo meager sets if and only if it is either meager or comeager. Additional equivalent conditions are given if $X=\mathbb{R}$. Second, we prove that if $A_1,\ldots,A_k\subseteq \mathbb{R}$ are subsets with the Baire property, $α_1,\ldots,α_k$ are nonzero reals with distinct finite sums, and $(x_{n,i}: n\in ω)$ are injective real sequences which converge to $0$ for each $i=1,\ldots,k$, then $$ \sum_{i=1}^k α_i(1_{A_i+x_{n,i}}-1_{A_i})=0 $$ modulo meager sets for all $n \in ω$ if and only if each $A_i$ is either meager or comeager. Finally, we provide some additional results in the case where the $α_1,\ldots,α_k$ do not have distinct finite sums. For instance, if $k=2$ and $α_1=α_2$, then the above equality holds modulo meager sets for all $n \in ω$ if and only if each $A_i$ is either meager or comeager or if $\{A_1,A_2\}$ is a partition of $\mathbb{R}$ modulo meager sets. Additional characterizations are given in the case $k\ge 3$. These results yield the category analogues of several results by Fejzić, Freiling, and Rinne in [J. London Math. Soc.~\textbf{82} (2010), no. 3, 717--732].
