*Journal of Combinatorial Designs*, **10** (2002), 111-115.

A cap in affine space AG(k,q) is a set A of k-tuples in
GF(q) such that whenever a_{1},a_{2},a_{3} are different elements of A
and
l_{i} in GF(q),l_{i} not 0, i=1,2,3 such that
l_{1}+l_{2}+l_{3}=0
we have
a_{1}l_{1}+a_{2}l_{2}+a_{3}l_{3} is not 0.

Denote by C_{k}(q) the maximum cardinality of a cap in AG(k,q), and
c_{k}(q)=C_{k}(q)/q^{k}. Clearly c_{k}(2)=1. Henceforth we assume q› 2.
The values C_{k}(q) for k‹ 4 are well-known.
Aside of these only a small number of
values are known:
C_{4}(3)=20 and C_{5}(3)=45.

Clearly C_{k}(q)‹= qC_{k-1}(q), hence c_{k}(q)‹= c_{k-1}(q). Our main
results may be seen as lower bounds on c_{k-1}(q)-c_{k}(q).
Meshulam proves an upper bound on the size of subsets
of abelian groups of odd order, which do not contain 3-term arithmetic
progressions.
A more careful analysis of Meshulam's method in the case of caps
shows that it can be generalized to cover also the characteristic 2 case.
Moreover stronger bounds can be obtained.

Download the preprint as pdf.

| home | List of publications |