Query Complexity of Sampling and Small Geometric Partitions

In this paper we study the following problem: Discrete partitioning problem (DPP): Let FqP denote the n-dimensional finite projective space over Fq. For positive integer k ≤ n, let {A}i=1 be a partition of (FqP ) such that 1. for all i ≤ N , A = ∏k j=1 A i j (partition into product sets), 2. for all i ≤ N , there is a (k − 1)-dimensional subspace L ⊆ FqP such that A ⊆ (L). What is the minimum value of N as a function of q, n, k? We will be mainly interested in the case k = n. DPP arises in an approach that we propose for proving lower bounds for the query complexity of generating random points from convex bodies. It is also related to other partitioning problems in combinatorics and complexity theory. We conjecture an asymptotically optimal partition for DPP and show that it is optimal in two cases: When the dimension is low (k = n = 2) and when the factors of the parts are structured, namely factors of a part are close to being a subspace. These structured partitions arise naturally as partitions induced by query algorithms. Our problem does not seem to be directly amenable to previous techniques for partitioning lower bounds such as rank arguments, although rank arguments do lie at the core of our techniques.