Restricted Sumsets in Finite Nilpotent Groups

TL;DR

This paper establishes a lower bound on the size of restricted sumsets in finite nilpotent groups, extending classical additive combinatorics results to a broader algebraic setting.

Contribution

It introduces a new lower bound for restricted sumsets in finite nilpotent groups, generalizing known results from abelian groups.

Findings

Lower bound of sumset size: min{p(G), |A|+|B|-2}

Applicable to non-abelian finite nilpotent groups

Extends classical sumset inequalities to a broader class of groups

Abstract

Suppose that $A,B$ are two non-empty subsets of the finite nilpotent group $G$. If $A\not=B$, then the cardinality of the restricted sumset $$A\dotplus B={a+b: a\in A, b\in B, a\neq b} $$ is at least $$\min{p(G),|A|+|B|-2},$$ where $p(G)$ denotes the least prime factor of $|G|$.