Exceptional set estimates in finite fields

Abstract: We study the exceptional set estimate for projections in $\mathbb{F}_q^n$. For each $V\in G(k,\mathbb{F}^n_q)$, 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 $t>n-2$.