Query complexity of sampling and small geometric partitions

TL;DR

This paper investigates the minimal number of product-structured partitions needed to cover certain subsets of finite projective spaces, focusing on the case where the partition dimension equals the space dimension.

Contribution

It introduces the discrete partitioning problem in finite projective spaces and analyzes the query complexity of sampling and partitioning in this geometric setting.

Findings

Derived bounds on the minimum number of partitions N

Characterized the structure of partitions in finite projective spaces

Provided insights into the complexity of geometric sampling problems

Abstract

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 $k=n$.