  In this paper we study the following problem:
  Discrete partitioning problem (DPP): Let $\mathbb{F}_q P^n$ denote the
$n$-dimensional finite projective space over $\mathbb{F}_q$. For positive
integer $k \leq n$, let $\{ A^i\}_{i=1}^N$ be a partition of $(\mathbb{F}_q
P^n)^k$ such that
  (1) for all $i \leq N$, $A^i = \prod_{j=1}^k A^i_j$ (partition into product
sets),
  (2) for all $i \leq N$, there is a $(k-1)$-dimensional subspace $L^i
\subseteq \mathbb{F}_q P^n$ such that $A^i \subseteq (L^i)^k$.
  What is the minimum value of $N$ as a function of $q,n,k$? We will be mainly
interested in the case $k=n$.
