Ja n 20 09 Large zero - free subsets of Z / p Z
نویسندگان
چکیده
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.
منابع مشابه
Proceedings of Integers Conference 2009 LARGE ZERO - FREE SUBSETS OF Z / p Z Jean - Marc Deshouillers Institut
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 cardinalities of which are close to the largest possible. In particular, we determine the cardinality of the largest zero-free subset of Z/pZ, when p is a sufficiently large p...
متن کاملRefinement of an Inequality Of
We prove: Let P(z) = P n k=0 a k z k be a complex polynomial with n 1 and a 0 an 6 = 0. If z is a zero of P, then we have for all real numbers t > 0: (*) jzj > ja 0 jt ja 0 j + Kn(t) with Kn(t) = 1 1 ? n(t) n min 1mn h (n(t) m ? n(t) n) max mpn Ap(t) + (1 ? n(t) m) max 1pn Ap(t) i ; n(t) = ja 0 j ja 0 j + max 1pn Ap(t) ; Ap(t) = 1 p p X k=1 ja k jt k : Inequality (*) sharpens a result of E. Lan...
متن کاملExpected Norms of Zero-One Polynomials
Let An = ̆ a0 + a1z + · · · + an−1z : a j ∈ {0, 1} ̄ , whose elements are called zeroone polynomials and correspond naturally to the 2n subsets of [n] := {0, 1, . . . , n − 1}. We also let An,m = {α(z) ∈ An : α(1) = m}, whose elements correspond to the `n m ́ subsets of [n] of size m, and let Bn = An+1 \ An, whose elements are the zero-one polynomials of degree exactly n. Many researchers have stu...
متن کاملJa n 20 09 Brody curves omitting hyperplanes
A Brody curve, a.k.a. normal curve, is a holomorphic map f from the complex line C to the complex projective space P such that the family of its translations {z 7→ f(z + a) : a ∈ C} is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n =
متن کاملCenter--like subsets in rings with derivations or epimorphisms
We introduce center-like subsets Z*(R,f), Z**(R,f) and Z1(R,f), where R is a ring and f is a map from R to R. For f a derivation or a non-identity epimorphism and R a suitably-chosen prime or semiprime ring, we prove that these sets coincide with the center of R.
متن کامل