On translations of subsets of the real line (Q2781355)

In this paper the authors discuss various questions, originating from \textit{W. Sierpiński}'s work [Fundam. Math. 19, 22-28 (1932; Zbl 0005.19701) and Fundam. Math. 35, 159-164 (1948; Zbl 0031.20506)], regarding translations of the real line and of other groups. E.g., Sierpiński proved that under the continuum hypothesis a nonempty proper subset of \(\mathbb{R}\) which is either Lebesgue measurable or has the Baire property has continuum many distinct translations: in this paper this theorem is proved without additional set-theoretic assumptions. A similar result is proved for subsets of the dyadic Cantor group \(2^\omega\), but in this case there could also be \(2^n\) distinct translations for some \(n>0\). NEWLINENEWLINENEWLINEThe key notion of the paper is that of \(J\)-almost invariant set, where \(J\) is a translation-invariant ideal of subsets of an additive group \(G\). In this setting the authors define \(A \subseteq G\) to be \(J\)-almost invariant if \(A + g \bigtriangleup A \in J\) for every \(g \in G\). \(J\)-almost invariant sets form a field of subsets of \(G\) and the authors provide other equivalent characterizations for this notion and a sufficient condition on \(J\) ensuring that the only \(J\)-almost invariant sets are the elements of \(J\) and their complements (these are the trivial \(J\)-almost invariant sets). NEWLINENEWLINENEWLINEWhen \(J\) is the Fréchet ideal of all subsets of \(G\) of cardinality less than \(|G|\), it is proved that if the coanalytic sets have the perfect set property then all analytic \(J\)-almost invariant subsets of \(\mathbb{R}\) are trivial. The paper includes also the proof of a theorem of A. Miller showing that \(V=L\) implies the existence of nontrivial \(J\)-almost invariant coanalytic subsets of \(\mathbb{R}\) and \(2^\omega\). The authors also show that this kind of results fail for countable groups. NEWLINENEWLINENEWLINEUnder Martin's Axiom the authors prove the existence of a subset of \(\mathbb{R}\) which is nontrivial almost invariant with respect to both Lebesgue-null and meager sets. They show that this result fails in ZFC alone by constructing a model where every almost invariant subset with respect to measure is trivial and another where every almost invariant subset with respect to category is trivial. It is apparently open whether it is consistent with ZFC that every almost invariant subset with respect to measure is trivial and the same holds for category.