Additive bases in abelian groups

Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B₁,⋯, B_k are generating subsets of G. We prove that if k > 2m ln log₂ |G|, then the multiset union B₁ ∪ B_k forms an additive basis of G; that is, for every g ∈ G, there exist A₁ ⊆ B₁,⋯, A_k ⊆ B_k such that g=∑^k _i=1∑_aεAi a. This generalizes a result of Alon, Linial and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where B₁,⋯, B_k are finite subsets of a vector space, we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form ∑ ^k_i=1∑_aεAi a, where A_i vary over all subsets of B_i for each i = 1,⋯, k. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.