tailieunhanh - Báo cáo toán học: "RESTRICTED SET ADDITION IN GROUPS, II. ˝ A GENERALIZATION OF THE ERDOS-HEILBRONN CONJECTURE"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: RESTRICTED SET ADDITION IN GROUPS, II. ˝ A GENERALIZATION OF THE ERDOS-HEILBRONN CONJECTURE. | RESTRICTED SET ADDITION IN GROUPS II. A GENERALIZATION OF THE ERDOS-HEILBRONN CONJECTURE Vsevolod F. Lev Institute of Mathematics Hebrew University Jerusalem 91904 Israel seva@ Submitted June 18 1998 Accepted January 29 2000 Abstract. In 1980 Erdos and Heilbronn posed the problem of estimating from below the number of sums a b where a 2 A and b 2 B range over given sets A B c Z pZ of residues modulo a prime p so that a b. A solution was given in 1994 by Dias da Silva and Hamidoune. In 1995 Alon Nathanson and Ruzsa developed a polynomial method that allows one to handle restrictions of the type f a b 0 where f is a polynomial in two variables over Z pZ. In this paper we consider restricting conditions of general type and investigate groups distinct from Z pZ. In particular for A B c Z pZ and R c A X B of given cardinalities we give a sharp estimate for the number of distinct sums a b with a b 2 R and we obtain a partial generalization of this estimate for arbitrary Abelian groups. 1. Background mapping restrictions For two subsets A and B of the set of elements of a group G we write A T B fa T b a 2 A b 2 B a bg. The group G Z pZ of residues modulo a prime p was historically first to emerge in this context hence the additive notation. In other words A T B is the set of all elements of G representable as a sum of two distinct elements from A and B. The Erdos-Heilbronn conjecture see 5 p. 95 resolved affirmatively in 4 cf. also 1 2 is the following. 1991 Mathematics Subject Classification. Primary 11B75 Secondary 11P99 05C25 05C35. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas sponsored by the Minerva Foundation Germany . 1 2 THE ELECTRONIC JOURNAL OF COMBINATORICS 7 2000 R4 Conjecture 1 Erdos and Heilbronn . For any two sets A B c ZfpZ 1 A B min A B 3 p . The set A B is obtained from A B a b a 2 A b 2 B by excluding those sums with b a. It seems plausible that 1 remains valid even if the sums to