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

作者：Balandraud Eric^*

来源：Israel Journal of Mathematics, 2012, 188(1): 405-429.

DOI：10.1007/s11856-011-0171-9

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 boolean AND (-A) = empty set in Z/pZ:
|Sigma(A)| >= min {p, 1 + |A|(|A| + 1)/2}.
The proof is similar in nature to Alon, Nathanson and Ruzsa's proof of the Erdos-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune in 1994). 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. As an 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. A generalization of this new addition theorem to the set of subsums of a sequence is also derived, leading to a structural result on zero-sum free sequences.

• 出版日期2012-3