Lower bounds for sumsets of multisets in Z_p²

2011/07/21 by Greg Martin, Martin, Greg, Alexis Peilloux +3
Mathematics · #11B13 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B13

paper · pdf · doi:10.48550/arxiv.1107.4392

13 pages. The quantitative bound in Theorem 1.8 has been improved, and a new coauthor has been added. These statements are not unrelated

arxiv created 2012/08/31 · arxiv updated 2012/09/03

Abstract

The classical Cauchy-Davenport theorem implies the lower bound n+1 for the number of distinct subsums that can be formed from a sequence of n elements of the cyclic group Z_p (when p is prime and n<p). We generalize this theorem to a conjecture for the minimum number of distinct subsums that can be formed from elements of a multiset in (Z_p)^m; the conjecture is expected to be valid for multisets that are not "wasteful" by having too many elements in nontrivial subgroups. We prove this conjecture in (Z_p)² for multisets of size p+k, when k is not too large in terms of p.

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