A Note on Combinatorial Derivation

Abstract

Given an infinite group $G$ and a subset $A$ of $G$ we let $\Delta (A)=g\in G:|gA\cap A|=\infty$ (this is sometimes called the combinatorial derivation of $A$). A subset $A$ of $G$ is called large if there exists a finite subset $F$ of $G$ such that $FA=G$. We show that given a large set $X$, and a decomposition $X=A_1\cup ...\cup A_n$, there must exist an $i$ such that $\Delta (A_i)$ is large. This answers a question of Protasov. We also answer a number of related questions of Protasov.

5 pages, fixed error in statement of Theorem 1

Cited by in corpus (4)

• On the subset Combinatorics of G-spaces
• Densities, submeasures and partitions of groups
• Relative size of subsets of a semigroup
• A note on the combinatorial derivation of non-small sets