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

by Jack Huizenga

Datum	Value	Source
TITLE	The minimum size of complete caps in (Z/nZ) 2	SVM HeaderParse 0.2
AUTHOR NAME	Jack Huizenga	SVM HeaderParse 0.2
AUTHOR AFFIL	Department of Mathematics; University of Chicago	SVM HeaderParse 0.2
AUTHOR ADDR	Chicago, IL 60637 USA	SVM HeaderParse 0.2
ABSTRACT	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, weprovethat max{4, √ 2p + 1}≤Φ(n) ≤ max{4,p+1}. 2 We conclude the paper with a large number of open problems in this area. 1	SVM HeaderParse 0.2
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