Some improved bounds on the number of directions not determined by a point set in the affine space AG(
k,
q) are presented. More precisely, if there are more than
p
e
(
q − 1) directions not determined by a set of
q
k-1 points
S{\mathcal S} then every hyperplane meets
S{\mathcal S} in 0 modulo
p
e+1 points. This bound is shown to be tight in the case
p
e
=
q
s
and when
q =
p
es
sets of
q
k-1 points that do not meet every hyperplane in 0 modulo
p
e+1 points and have a little less than
p
e
(
q − 1) non-determined directions are constructed.
Keywords Directions determined by a function - Functions over finite fields - Ovoids
AMS Classifications 51E99 - 11T06
The author acknowledges the support of the Ramon y Cajal programme and the project MTM2005-08990-C02-01 of the Spanish Ministry
of Science and Education and the project 2005SGR00256 of the Catalan Research Council.