Structure of sets with small product sets in torsion-free groups, cyclic groups of prime orders and abelian groups

TL;DR

This paper establishes optimal lower bounds for generalized product sets in torsion-free and cyclic groups, characterizes the structure of sets achieving these bounds, and extends inverse theorems in additive combinatorics.

Contribution

It provides the first optimal bounds and structural characterizations for generalized product sets in non-abelian torsion-free groups and cyclic groups of prime order, extending classical additive combinatorics results.

Findings

Optimal lower bounds for generalized product sets in torsion-free groups.

Structural characterization of sets attaining the bounds.

New proofs and extensions of the DeVos-Goddyn-Mohar Theorem and subsequence sum results.

Abstract

Let $\ell$ and $m$ be positive integers with $\ell \leq m$, and let $\mathcal{A} = (A_1, \ldots, A_m)$ be a finite sequence of finite subsets of a group $G$ (not necessarily abelian), written multiplicatively. The {\it generalized product set} $\Pi^{\ell}(\mathcal{A})$ is the set of all elements of $G$ which can be represented as a product of exactly $\ell$ elements from $\ell$ distinct sets from $\mathcal{A}$ taken in any order. DeVos, Goddyn and Mohar obtained the nontrivial lower bound for the size of this product set when $G$ is abelian. The DeVos-Goddyn-Mohar Theorem is a fundamental result in additive combinatorics which unifies various results from zero-sum combinatorics and has connections with subsequence sums and sumsets. In this paper, we obtain an optimal lower bound for the size of generalized product set ${\Pi}^{\ell}(\mathcal{A})$ in torsion-free groups (not necessarily abelian), and characterize the structure of underlying sets in the sequence $\mathcal{A} = (A_1, \ldots, A_m)$ for which ${\Pi}^{\ell}(\mathcal{A})$ achieves the optimal lower bound. By slightly modifying the arguments of the proofs in the case of torsion-free groups, we derive such inverse theorems in cyclic groups of prime orders also. Our proof of these result also yields a new proof of DeVos-Goddyn-Mohar Theorem in $\mathbb{Z}_p$. Moreover, we extend these inverse results to arbitrary abelian groups. Furthermore, as an application, we generalize a theorem for subsequence sums due to Hamidoune in torsion-free groups, and obtain several other results for subsequence sums in arbitrary groups.