Ja n 20 09 Large zero - free subsets of Z / p Z

A finite subset A of an abelian group G is said to be zero-free if the identity element of G cannot be written as a sum of distinct elements from A. In this article we study the structure of zero-free subsets of Z/pZ the cardinality of which is close to largest possible. In particular, we determine the cardinality of the largest zero-free subset of Z/pZ, when p is a sufficiently large prime. For a finite abelian group (G, +) and a subset A of G, we set A♯ = {∑b∈B b : B ⊂ A, B 6 = ∅ }. We say A is zero-free if 0 / ∈ A♯; in other words A is zero-free if 0 can not be expressed as a sum of distinct elements of A. In 1964, Erdős and Heilbronn [5] made the following conjecture, supported by examples showing that the upper bound they conjectured is, if correct, very close to being best possible. Conjecture 1. Let A be a subset of Z/pZ. If A is zero-free, we have Card(A) ≤ √2p. Up to recently, the best result concerning zero-free subsets of Z/pZ was that of Hamidoune and Zémor [3] who proved in 1996 that their cardinality is at most √ 2p + 5 ln p, thus showing that the constant √ 2 in the above conjecture is sharp. The study of this question has been revived more recently. Freiman and the first named author introduced a method based on trigonometrical sums which led to the description of large incomplete subsets [2] as well as that of large zero-free subsets [1] of Z/pZ. Recall that a subset A of G is said to be incomplete if A♯ ∪ {0} is not equal to G. Szemerédi and Van Vu [6], as a consequence of their result on long arithmetic progressions in sumsets, gave structure results for zero-free subsets leading to the optimal bound for the total number of such subsets of Z/pZ. As it was noticed independently by Nguyen, Szemerédi and Van Vu [4] on one side and us on the other one, both methods readily lead to a proof of the Erdős-Heilbronn conjecture for zero-free subsets1. The aim of the present paper is to study the description of rather large zero-free subsets of Z/pZ. We start by reviewing the present knowledge on zero-free subsets of Z/pZ. Notation 2. We denote by σp the canonical homomorphism from Z onto Z/pZ; for an element a in Z/pZ, we denote by ā be the integer in (− 2 , p 2 ] such that a = σp(ā) and let |a|p = |ā|. Given a set A ⊂ Z/pZ, we denote by Ā the set {ā : a ∈ A}. For d ∈ Z/pZ, we write d · A := {da : a ∈ A}. Given any real numbers x, y with x ≤ y, we write [x, y]p Van H. Vu and the first named author exchanged this information during a private conversation held in Spring 2006.