A note on the combinatorial derivation of non-small sets

TL;DR

This paper proves that in infinite groups, every non-small subset is necessarily elta-large, providing an answer to a question posed by Protasov about the structure of such sets.

Contribution

It establishes that all non-small sets in infinite groups are elta-large, clarifying the relationship between smallness and elta-largeness.

Findings

Every non-small set is elta-large in infinite groups.

Answers Protasov's question on the structure of non-small sets.

Provides a combinatorial perspective on set largeness in groups.

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 \emph{combinatorial derivation} of $A$). A subset $A$ of $G$ is called: \emph{large} if there exists a finite subset $F$ of $G$ such that $FA=G$; \emph{$\Delta$-large} if $\Delta(A)$ is large and \emph{small} if for every large subset $L$ of $G$, $(G \setminus A) \cap L$ is large. In this note we show that every non-small set is $\Delta$-large, answering a question of Protasov.