On the volume set of point sets in vector spaces over finite fields

2009/03/13 by Vinh, Le Anh
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.0903.2510

Abstract

We show that if E is a subset of the d-dimensional vector space over a finite field \mathbbmF_q (d ≥ 3) of cardinality |E| ≥ (d-1)q^{d - 1}, then the set of volumes of d-dimensional parallelepipeds determined by E covers \mathbbmF_q. This bound is sharp up to a factor of (d-1) as taking E to be a (d - 1)-hyperplane through the origin shows.

Related

• Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields
• A Furstenberg-Katznelson-Weiss type theorem on (d + 1)-point configurations in sets of positive density in finite field geometries
• Volumes spanned by k point configurations in ℝ^d
• VC-Dimension of Hyperplanes over Finite Fields
• Ubiquity of simplices in subsets of vector spaces over finite fields