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

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} $Π^{\ell}(\mathcal{A})$ 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 $Π^{\ell}(\mathcal{A})$ 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 $Π^{\ell}(\mathcal{A})$ 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.