A variant of Kemnitz conjecture

摘要

For any integer n greater than or equal to 3, by g(Z(n) circle plus Z(n)) we denote the smallest positive integer t such that every subset of cardinality t of the group Z(n) circle plus Z(n) contains a subset of cardinality n whose sum is zero. Kemnitz (Extremalprobleme fur Gitterpunkte, Ph.D. Thesis, Technische Universitat Braunschweig, 1982) proved that g(Z(p) circle plus Z(p)) = 2p - 1 for p = 3, 5, 7. In this paper, as our main result, we prove that g(Z(p) circle plus Z(p)) = 2p - 1 for all primes p greater than or equal to 67. (C) 2004 Elsevier Inc. All rights reserved.