On (<i>k</i>, <i>l</i>)-sets in cyclic groups of odd prime order

Abstract

Let A be a finite Abelian group written additively. For two positive integers k , l with k ≠ l , we say that a subset S ⊂ A is of type ( k , l ) or is a ( k , l ) - set if the equation x 1 + x 2 + … + x k − x k+1 −… − x k+1 = 0 has no solution in the set S . In this paper we determine the largest possible cardinality of a ( k , l )-set of the cyclic group ℤ P where p is an odd prime. We also determine the number of ( k , l )-sets of ℤ p which are in arithmetic progression and have maximum cardinality.