Kaleidoscopical Configurations in G-spaces

TL;DR

This paper introduces and constructs kaleidoscopical configurations in various G-spaces, including finite Abelian groups and Euclidean spaces, providing new methods and characterizations for these colorings.

Contribution

It develops new constructions and techniques for kaleidoscopical configurations in G-spaces, including a splitting construction and characterizations in finite Abelian groups and ultrametric spaces.

Findings

Constructed kaleidoscopical configurations in arbitrary G-spaces.

Reduced the problem to factorization in finite Abelian groups.

Described all configurations in ultrametric spaces and Euclidean spaces.

Abstract

Let $G$ be a group and $X$ be a $G$-space. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a surjective coloring $\chi:X\to Y$ such that the restriction of $\chi$ on each subset $gF$, $g\in G$ is a bijection. We give some constructions of kaleidoscopical configurations in an arbitrary $G$-space, develop some kaleidoscopical technique for Abelian groups (considered as $G$-spaces with the action $(g,x)\mapsto g+x$), and describe kaleidoscopical configurations in the cyclic groups of order $N=p^m$ or $N=p_1... p_k$ where $p$ is prime and $p_1,...,p_k$ are distinct primes. Let $G$ be a group and $X$ be a $G$-space. 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 $2^c$ (unsplittable) kaleidoscopical configurations of cardinality continuum in the Euclidean space $R^n$.