Exceptional set estimates in finite fields

We study the exceptional set estimate for projections in \(F_q^n\). For each \(V∈ G(k,F^n_q)\), let \(π_V F_q^n→ V\) be the projection map. We prove the following result: If \(A⊂ F_q^n\) with \(#A=q^a\) (\(n-1≤ a≤ n\)) and \(0< s<a+n-2/2\), then \(# \{V∈ G(n-1,F^n_q) #π_V(A)< q^s \} qⁿ⁻²\). This improves the previous range \(0<s<n-1/na\). Also, our range of \(s\) is sharp in the sense that if \(s>a+n-2/2\), then the right hand side above should be at least \(q^t\) for some \(t>n-2\).