Let L be a set of lines of an affine space over a field and let S be a
set of points with the property that every line of L is incident with at
least N points of S. Let D be the set of directions of the lines of L
considered as points of the projective space at infinity. We give a geometric
construction of a set of lines L, where D contains an Nn−1 grid and
where S has size 2((1/2)N)n, given a starting configuration in the plane.
We provide examples of such starting configurations for the reals and for
finite fields. Following Dvir's proof of the finite field Kakeya conjecture and
the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a
lower bound on the size of S dependent on the ideal generated by the
homogeneous polynomials vanishing on D. This bound is maximised as
((1/2)N)n plus smaller order terms, for n⩾4, when D contains
the points of a Nn−1 grid.Comment: A few minor changes to previous versio