## Abstract

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} ∪ B_{k} forms an additive basis of G; that is, for every g ∈ G, there exist A_{1} ⊆ B_{1},⋯, 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 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 ∑ ^{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.

Original language | English |
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Pages (from-to) | 799-809 |

Number of pages | 11 |

Journal | International Journal of Number Theory |

Volume | 6 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jun 2010 |

Externally published | Yes |

## Keywords

- Additive bases
- additive combinatorics