ADDITIVE BASES IN ABELIAN GROUPS

作者：Lev Vsevolod F^*; Muzychuk Mikhail E; Pinchasi Rom

来源：International Journal of Number Theory, 2010, 6(4): 799-809.

DOI：10.1142/S1793042110003216

• Additive bases
• additive combinatorics

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 vertical bar G vertical bar, then the multiset union B(1) boolean OR ... B(k) forms an additive basis of G; that is, for every g is an element of G, there exist A(1) subset of B(1),..., A(k) subset of B(k) such that g = Sigma(k)(i=1)Sigma(a subset of Ai) 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 Sigma(k)(i=1)Sigma(a is an element of Ai) 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.

• 出版日期2010-6