Generalized point configurations in \mathbb F_q^d

2023/08/21 by Bright, Paige, Fang, Xinyu, Heritage, Barrett +4
#52C10 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.10853

Abstract

In this paper, we generalize \citeIosevichParshall, \citeLongPaths and \citecycles by allowing the distance between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of \F_q^d for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.

Related

• Embedding distance graphs in finite field vector spaces
• Long paths in the distance graph over large subsets of vector spaces over finite fields
• Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness
• Interpolation of point configurations in the discrete plane
• Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields