Lower Bounds for Sumsets of Multisets in Z2

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 m p ; 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 2 p for multisets of size p + k, when k is not too large in terms of p.Lemma 1.1 (Cauchy-Davenport Theorem). Let A and B be subsets of Z p , and define A + B to be the set of all elements of the formThe lower bound is easily seen to be best possible by taking A and B to be intervals, for example. It is also easy to see that the lower bound of #A + #B − 1 does not hold for general abelian groups G (take A and B to be the same nontrivial subgroup of G). There is, however, a wellknown generalization obtained by Kneser in 1953 [4], which we state in a slightly simplified form that will be quite useful for our purposes (see [7, Theorem 4.1] for an elementary proof): Lemma 1.2 (Kneser's Theorem). Let A and B be subsets of a finite abelian group G, and let m be the largest cardinality of a proper subgroup of G. Then #(A + B) ≥ min{#G, #A + #B − m}.Given a sequence A = (a 1 , . . . , a k ) of (not necessarily distinct) elements of an abelian group G, a related result involves its sumset ΣA, which is the set of all sums of any number of elements chosen from A (not to be confused with A + A, which it contains but usually properly): ΣA = j∈J a j : J ⊆ {1, . . . , k} .(Note that we allow J to be empty, so that the group's identity element is always an element of ΣA.) When G = Z p , one can prove the following result by writing ΣA = {0, a 1 } + · · · + {0, a k } and applying the Cauchy-Davenport theorem inductively: Lemma 1.3. Let A = (a 1 , . . . , a k ) be a sequence of nonzero elements of Z p . Then #ΣA ≥ min{p, k + 1}.

On Sumsets of Multisets in $\mathbb{Z}_p^m$

“…Hence we can concentrate on the case p |A| 2p−2. Martin, Peilloux and Wong [3] have made the following conjecture.…”

“…If true, this conjecture would be sharp as pointed out in [3]: First, for k p − 3, the multiset A may consist of p − 1 copies of (1, 0) and k + 1 copies of (0, 1), so that A = Z p × {0, . .…”

“…Second, for k = p − 2, A may consist of p − 2 copies of (1, 0) and one copy of each (i, 1), 0 i p − 1, so that A = Z 2 p \ {(p − 1, 0)}. Martin, Peilloux and Wong [3] proved the conjecture when…”

“…Here we are interested in the relationship between |A| and # A. As pointed out for instance in [3,Lemma 1.3], in case G = Z p one gets the following result easily by multiple applications of the Cauchy-Davenport inequality (see [6,Theorem 5.4]). Lemma 1.…”

“…In particular it is "wasteful" for A to contain more than p−1 elements from any subgroup since by Lemma 1 already p − 1 elements guarantee that A contains the whole subgroup. In light of this we make the following definition (following [3]).…”