The minimum size of complete caps in (ℤ/nℤ)2

A line in (ℤ/nℤ)² is any translate of a cyclic subgroup of order n. A subset X ⊂ (ℤ/nℤ)² is a cap if no three of its points are collinear, and X is complete if it is not properly contained in another cap. We determine bounds on Φ(n), the minimum size of a complete cap in (ℤ/nℤ)². The other natural extremal question of determining the maximum size of a cap in (ℤ/nℤ) ² is considered in [8]. These questions are closely related to well-studied questions in finite affine and projective geometry. If p is the smallest prime divisor of n, we prove that max{4, √2p + 1/2 } ≤ Φ(n) ≤ max{4, p + 1}. We conclude the paper with a large number of open problems in this area.