Linear relations modulo meager sets

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 ∈ 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=R. Second, we prove that if A₁,…,Aₖ⊆ R are subsets with the Baire property, α₁,…,αₖ are nonzero reals with distinct finite sums, and (xn,i: n∈ ω) are injective real sequences which converge to $0$ for each i=1,…,k, then ∑ᵢ₌₁ᵏ αᵢ(1_Aᵢ+xn,i-1Aᵢ)=0 modulo meager sets for all n ∈ ω if and only if each Aᵢ is either meager or comeager. Finally, we provide some additional results in the case where the α₁,…,αₖ do not have distinct finite sums. For instance, if $k=2$ and α₁=α₂, then the above equality holds modulo meager sets for all n ∈ ω if and only if each Aᵢ is either meager or comeager or if ₁,A₂\ is a partition of R modulo meager sets. Additional characterizations are given in the case k≥ 3. These results yield the category analogues of several results by Fejzić, Freiling, and Rinne in [J. London Math. Soc.~82 (2010), no. 3, 717--732].