Lower bounds for sumsets of multisets in Z_p^2

TL;DR

This paper extends the classical Cauchy-Davenport theorem to multisets in (Z_p)^2, providing lower bounds on the number of distinct subsums and proving the conjecture for certain multiset sizes.

Contribution

It generalizes a fundamental additive combinatorics result to higher dimensions and multisets, and proves the conjecture in (Z_p)^2 for specific multiset sizes.

Findings

Established lower bounds for sumsets of multisets in (Z_p)^2.

Proved the conjecture for multisets of size p+k with small k.

Extended classical sumset results to multiset scenarios in higher dimensions.

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.