We prove that for every k ∈ N each countable infinite group G admits a partition G = A ∪ B into two sets which are k-meager in the sense that for every k-element subset K ⊂ G the sets KA and KB are not thick. The proof is based on the fact that G possesses a syndetic submeasure, i.e., a left-invariant submeasure µ : P(G) → [0, 1] such that for each ε > 1
Abstract. Given a G-space X and a non-trivial G-invariant ideal I of subsets of X, we prove that for every partition X = A 1 ∪ · · · ∪ An of X into n ≥ 2 pieces there is a piece A i of the partition and a finite set F ⊂ G of cardinality |F | ≤ φ(n + 1) := max 1
Let $G$ be a group and $X$ be a $G$-space with the action $G\times X\rightarrow X$, $(g,x)\mapsto gx$. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a coloring $\chi:X\rightarrow C$ such that the restriction of $\chi$ on each subset $gF$, $g\in G$, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary $G$-space, reduce the problem of characterization of kaleidoscopical configurations in a finite Abelian group $G$ to a factorization of $G$ into two subsets, and describe all kaleidoscopical configurations in isometrically homogeneous ultrametric spaces with finite distance scale. Also we construct $2^{\mathfrak c}$ (unsplittable) kaleidoscopical configurations of cardinality $\mathfrak c$ in the Euclidean space $\mathbb{R}^n$.
We define the scattered subsets of a group as asymptotic counterparts of scattered subspaces of a topological space, and prove that a subset A of a group G is scattered if and only if A contains no piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that µ(A) = 0 for each left invariant Banach measure µ on G.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.