On the maximum size of a k-zero-free set in an abelian group

Abstract

A subset A of elements in an abelian group G is called k-zero-free if the equation x1 + x2 + ... + xk = 0 has no solution in A. A k-zero-free set A in G is called maximal if A ∪ {x} is k-zero-free for no x ∈ G\A. Some bounds for the maximum size of a k-zero free set are obtained. In particular, we determine the maximum speed of a k-zero-free arithmetic progression in the cyclic group Zn and find the upper and lower bounds for the maximum size of a k-zero-free set in an abelian group G. We describe the structure of a maximal k-zero-free set A in the cyclic group Zn provided that gcd(n, k) = 1 and k|A| ≥ n + 1.