2026/07/25 by Marek Balcerzak, Paolo Leonetti
#math.GN #math.CA
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 \A1,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].