Distances from points to planes

TL;DR

This paper establishes a lower bound on the number of distances from points to affine hyperplanes in finite fields, improving previous bounds in higher dimensions.

Contribution

It proves a new lower bound on the number of point-plane distances in finite fields, enhancing earlier results for dimensions three and above.

Findings

Lower bound of q/2 on distances when |E||F| > q^{d+1}

Significant improvement over previous exponents in dimensions ≥ 3

Applicable to finite field geometries with affine hyperplanes

Abstract

We prove that if $E \subset {\Bbb F}_q^d$, $d \ge 2$, $F \subset \operatorname{Graff}(d-1,d)$, the set of affine $d-1$-dimensional planes in ${\Bbb F}_q^d$, then $|\Delta(E,F)| \ge \frac{q}{2}$ if $|E||F|>q^{d+1}$, where $\Delta(E,F)$ the set of distances from points in $E$ to lines in $F$. In dimension three and higher this significantly improves the exponent obtained by Pham, Phuong, Sang, Vinh and Valculescu.