Linear relations modulo meager sets

Abstract

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, $\alpha_1,\ldots,\alpha_k$ are nonzero reals with distinct finite sums, and $(x_{n,i}: n\in \omega)$ are injective real sequences which converge to $0$ for each $i=1,\ldots,k$, then $$ \sum_{i=1}^k \alpha_i(1_{A_i+x_{n,i}}-1_{A_i})=0 $$ modulo meager sets for all $n \in \omega$ if and only if each $A_i$ is either meager or comeager. Finally, we provide some additional results in the case where the $\alpha_1,\ldots,\alpha_k$ do not have distinct finite sums. For instance, if $k=2$ and $\alpha_1=\alpha_2$, then the above equality holds modulo meager sets for all $n \in \omega$ 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 Fejzi\'{c}, Freiling, and Rinne in [J. London Math. Soc.~\textbf{82} (2010), no. 3, 717--732].