Exceptional set estimates in finite fields

TL;DR

This paper investigates the size of exceptional sets of projections in finite fields, providing improved bounds and establishing the sharpness of these bounds for subsets of finite field vector spaces.

Contribution

It improves previous bounds on the exceptional set estimates for projections in finite fields and proves the sharpness of these bounds.

Findings

Improved the range for exceptional set estimates from 0<s<(n-1)/n * a to 0<s<(a+n-2)/2.

Established that the new bounds are sharp, with the exceptional set size at least q^t for some t>n-2 when s exceeds the threshold.

Provided precise quantitative bounds for the number of projections with small image sets in finite field vector spaces.

Abstract

We study the exceptional set estimate for projections in $\mathbb{F}_q^n$. For each $V\in G(k,\mathbb{F}^n_q)$, let $$ \pi_V: \mathbb{F}_q^n\rightarrow V $$ be the projection map. We prove the following result: If $A\subset \mathbb{F}_q^n$ with $\#A=q^a$ ($n-1\le a\le n$) and $0< s<\frac{a+n-2}{2}$, then $$ \# \{V\in G(n-1,\mathbb{F}^n_q): \#\pi_V(A)< q^s \}\lessapprox q^{n-2}.$$ This improves the previous range $0<s<\frac{n-1}{n}a$. Also, our range of $s$ is sharp in the sense that if $s>\frac{a+n-2}{2}$, then the right hand side above should be at least $q^t$ for some $t>n-2$.