  Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose
that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln
\log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive
basis of $G$; that is, for every $g\in G$ there exist $A_1\subset B_1, ...,
A_k\subset B_k$ such that $g=\sum_{i=1}^k\sum_{a\in A_i} a$. 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 $B_1, ...,
B_k$ 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 $\sum_{i=1}^k
\sum_{a\in A_i} a$, 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.
