Exceptional set estimates in finite fields

We study the exceptional set estimate for projections in \(\mathbb{F}_q^n\). For each \(V\in G(k,\mathbb{F}^n_q)\), let \(\pi_V\colon \mathbb{F}_q^n\rightarrow V\) 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 \(\# \{V\in G(n-1,\mathbb{F}^n_q)\colon \#\pi_V(A)< q^s \}\lessapprox q^{n-2}\). 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 \(t>n-2\).