Exceptional set estimate through Brascamp-Lieb inequality

Fix integers $1\le k<n$, and numbers $a,s$ satisfying $0<s<\min\{k,a\}$. The problem of exceptional set estimate is to determine $$T(a,s):=\sup_{A\subset \mathbb{R}^n\ \text{dim}A=a}\text{dim}(\{ V\in G(k,n): \text{dim}(\pi_V(A))<s \}).$$ In this paper, we prove a new upper bound for $T(a,s)$ by using Brascamp-Lieb inequality. As one of the corollary, we obtain the estimate $$T(a,\frac{k}{n}a)\le k(n-k)-\min\{k,n-k\},$$ which improves a previous result $T(a,\frac{k}{n}a)\le k(n-k)-1$ of He. By constructing examples, we can determine the explicit value of $T(a,s)$ for certain $(a,s)$: When $k\le \frac{n}{2}$, $\beta\in(0,1]$ and $\gamma\in(\beta,\frac{k}{n}(1+\beta)]$, we have $$T(1+\beta,\gamma)=k(n-k)-k.$$ When $k\ge \frac{n}{2}$, $\beta\in(0,1]$ and $\gamma\in (\beta, (1-\frac{k}{n})+\frac{k}{n}\beta]$, we have $$T(n-1+\beta,k-1+\gamma)=k(n-k)-(n-k).$$