Exceptional set estimates in finite fields

Abstract

We study the exceptional set estimate for projections in ${\mathbb{F}}_q^n$. For each $V\in G(k,{\mathbb{F}}_q^n)$, let $${\pi }_V:{\mathbb{F}}_q^n\to V$$ 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 $$\#\{V\in G(n-1,{\mathbb{F}}_q^n):\#{\pi }_V(A)<q^s\}⪅q^{n-2}.$$ 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 $t>n-2$.

16 pages; some modifications are made