Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B1,⋯, Bk are generating subsets of G. We prove that if k > 2m ln log2 |G|, then the multiset union B1 ∪ Bk forms an additive basis of G; that is, for every g ∈ G, there exist A1 ⊆ B1,⋯, Ak ⊆ Bk 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,⋯, Bk 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 ∑ ki=1∑aεAi a, where Ai vary over all subsets of Bi 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.
- Additive bases
- additive combinatorics