Three-fold Restricted Set Addition in Groups

Let G be a finite abelian group with a ‘sufficiently small’ proportion of elements of order two. We prove that for any subset A ⊆ G of cardinality | A | > 513| G | and any group elementg ∈ G, there exist pairwise distinct a1, a2,a3 ∈ A such that a1 + a2 + a3 = g— unless A is contained in a coset of an index two subgroup or in a union of two cosets of an index five subgroup. immediate corollary our result implies a recent conjecture of Gallardo et al. who addressed the groupG = Znwith n odd and sets of cardinality | A | > 25n.