On the Erd\H{o}s-Falconer distance problem for two sets of different size in vector spaces over finite fields

TL;DR

This paper investigates the Erdős-Falconer distance problem over finite fields for two sets of different sizes, providing results that match the conjectured order of magnitude within certain size ranges.

Contribution

It extends the finite field distance problem to two sets of different sizes and establishes results consistent with the conjecture in specific size ranges.

Findings

Results match the conjectured order of magnitude for certain set sizes.

Provides new bounds for the distance problem involving two differently sized sets.

Advances understanding of distance distributions in finite field vector spaces.

Abstract

We consider a finite fields version of the Erd\H{o}s-Falconer distance problem for two different sets. In a certain range for the sizes of the two sets we obtain results of the conjectured order of magnitude.