AN ADDITION THEOREM AND MAXIMAL ZERO-SUM FREE SETS IN Z/pZ

Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying A ∩ (−A) = ∅ in Z/pZ: |Σ(A)| > min  p, 1 + |A|(|A|+ 1) 2 ff . The proof is similar in nature to Alon, Nathanson and Ruzsa’s proof of the Erdös-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune [10]). A key point in the proof of this theorem is the evaluation of some binomial determinants that have been studied in the work of Gessel and Viennot. A generalization to the set of subsums of a sequence is derived, leading to a structural result on zero-sum free sequences. As another application, it is established that for any prime number p, a maximal zero-sum free set in Z/pZ has cardinality the greatest integer k such that k(k + 1) 2 < p, proving a conjecture of Selfridge from 1976. introduction Given two subsets A and B of an abelian group, we define their sumset: A+B = {a+ b|a ∈ A, b ∈ B}, we denote also a+B the sumset {a}+B. A first important addition theorem was discovered by Cauchy in 1813 and has been rediscovered a century later by Davenport: Theorem. (Cauchy-Davenport [5, 7, 8]) Let p be a prime number, A and B be two subsets of A ⊂ Z/pZ, then: |A+B| > min {p, |A|+ |B| − 1} . This theorem can easily be extended to the sumset of more than two sets: |∑Ai| > min {p, ∑ (|Ai| − 1) + 1}. Many proofs of the Cauchy-Davenport Theorem have been published and generalizations have been made in abelian groups or in torsion-free groups; Chowla’s Theorem [6], Mann’s Theorem [28], Kneser’s Theorem [23, 24, 4], see also [30]. Another topic in addition theory consists in investigating the cardinality of the restricted sumset: A+̇B = {a + b|a ∈ A, b ∈ B, a 6= b}. In 1964, Erdös and Heilbronn made a famous conjecture that became in 1994 the following theorem by Dias da Silva and Hamidoune: Theorem. (Dias da Silva, Hamidoune [10]) Let p be a prime number and A ⊂ Z/pZ. For a natural integer h, denote hA = A+̇ . . . +̇A } {{ } h times the set of subsums of h pairwise distinct elements of A. Then, |hA| > min{p, h(|A| − h) + 1}. In this article, we focus our interest on the set of all subsums: 1