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 $A\cap (-A)=\mathrm{\varnothing }$ in $\mathbb{Z}\mathrm{/}p\mathbb{Z}$: $$\mathrm{\mid }\mathrm{\Sigma }(A)\mathrm{\mid }⩾\min p,1+\frac{\mathrm{\mid }A\mathrm{\mid }(\mathrm{\mid }A\mathrm{\mid }+1)}{2}\mathrm{.}$$ The proof is similar in nature to Alon, Nathanson and Ruzsa’s proof of the Erd"os-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune \cite{DH}). 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 $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ has cardinality the greatest integer $k$ such that $$\frac{k(k+1)}{2}<p,$$ proving a conjecture of Selfridge from 1976.